Linear programming is a method used in mathematics to achieve the best possible outcome in a given situation—typically maximising or minimising a quantity such as profit, cost, or time—while working within a set of constraints. For A-Level Maths, linear programming appears in the Decision Mathematics module (also known as D1 or D2 in some exam boards), and is often presented through real-life scenarios involving resource allocation, budgeting, production limits, and transportation planning.
📝 Linear Programming Practice Questions
1. Defining decision variables (e.g. number of items produced)
2. Formulating a linear objective function to maximise or minimise
3. Writing a set of linear inequalities that represent constraints
4. Graphing the feasible region and identifying possible solutions
5. Using a corner-point or vertex-testing method to find the optimal solution
A transport company has two types of trucks, Type A and Type B. Type A has a refrigerated capacity of 
and a non-refrigerated capacity of 
. In contrast, Type B has the same overall volume with equal refrigerated and non-refrigerated stock sections. A grocer must hire trucks to transport 
of refrigerated stock and 
of non-refrigerated stock. The cost per kilometre of Type A is 
, and 
for Type B. How many trucks of each type should the grocer rent to achieve the minimum total cost?
a) Choose the unknowns.
x = Type A trucks
y = Type B trucks
b) Write the objective function.
c) Write the constraints as a system of inequalities.
d) Find the set of feasible solutions that graphically represent the constraints.
e) Calculate the coordinates of the vertices from the compound of feasible solutions.
f) Calculate the value of the objective function at each vertex to determine which has the maximum or minimum values.
As x and y must be natural numbers round the value of y.
By default, we see what takes the value x to y = 66 in the equation 
, which is within the feasible solutions.
The minimum cost is 
. To achieve this 51 trucks of Type A and 66 trucks of Type B are needed.
A school is preparing a trip for 400 students. The transportation company has 10 buses of 50 seats each and 8 buses of 40 seats but only has 9 drivers available. The rental cost for a large bus is 
and 
for a small bus. Calculate how many buses of each type should be used for the trip for the least possible cost.
a)Choose the unknowns.
x = small buses
y = big buses
b) Write the objective function.
c) Write the constraints as a system of inequalities.
d) Find the set of feasible solutions that graphically represent the constraints.
e) Calculate the coordinates of the vertices from the compound of feasible solutions.
f) Calculate the value of the objective function at each vertex to determine which has the maximum or minimum values.
Hence, the minimum cost is 
. This is achieved with 4 large and 5 small buses.
We substituted the points (0,9), (0,8), and (5,4) in the equation to determine the minimum cost. However, you can tell this by directly looking at the graph. The coordinate (5,4) comes under the feasible region and is the minimum point of it.
A store wants to liquidate 200 shirts and 100 pairs of pants from last season. They have decided to put together two offers, A and B. Offer A is a package of one shirt and a couple of pants which will sell for . Offer B is a package of three shirts and a pair of pants, which will sell for 
. The store does not want to sell less than 20 packages of Offer A and less than 10 of Offer B. How many boxes do they have to deal with to maximize the money generated from the promotion?
a) Choose the unknowns.
x = number of packages of Offer A
y = number of packages of Offer B
b) Write the objective function.
c) Write the constraints as a system of inequalities.
d) Find the set of feasible solutions that graphically represent the constraints.
Example 3 - part d
e) Calculate the coordinates of the vertices from the compound of feasible solutions.
Example 3 - part e
f) Calculate the value of the objective function at each vertices to determine which has the maximum or minimum values.
50 packages of each offer generate a maximum of 
in sales.
🔹 Steps for Solving Linear Programming Problems
To solve a linear programming problem effectively, follow these key steps. These are the standard methods used at A-Level for two-variable problems that can be solved graphically.
1. Define the decision variables
Start by identifying what the problem is asking you to find. Usually, this involves quantities such as the number of units produced or hours allocated. Assign variables (typically x and y) to represent these unknowns.
Example: Let x be the number of tables produced, and y the number of chairs.
2. Write the objective function
The objective function is the formula you want to maximise or minimise—often profit, cost, or time. It will usually be a linear equation in terms of x and y.
Example: Maximise profit
3. Formulate the constraints
Write out the inequalities based on the problem's limitations—such as time, materials, or budget. These are the conditions the solution must satisfy. Also, include non-negativity constraints x≥0x \geq 0x≥0 and y≥0y \geq 0y≥0, since negative quantities often don’t make sense in real-world scenarios.
Example:
(material limit)
(labour limit)
4. Draw the feasible region
On graph paper or using graphing software, plot each constraint as a straight line and shade the region that satisfies all the inequalities. The feasible region is the area where all constraints overlap.
5. Identify the corner (vertex) points
The maximum or minimum value of the objective function will occur at one of the corner points (vertices) of the feasible region. Find the coordinates of each vertex—either by observation, solving equations simultaneously, or using intersection points.
6. Evaluate the objective function at each vertex
Substitute the coordinates of each vertex into the objective function. Compare the results to determine the maximum or minimum value, depending on what the question asks.
Example:
Test
at each vertex and identify the one that gives the highest profit.
7. State the final solution clearly
Write out the optimal values of x and y, and what they represent in context (e.g. number of items to produce), along with the maximum or minimum value of the objective function.
These concepts combine algebra, graphical representation, and logical reasoning—making linear programming a valuable tool not only in mathematics exams but also in solving practical problems in business, science, and industry.
In the exercises below, you’ll find word problems typical of A-Level assessments, each followed by a fully worked solution. These will help you understand how to model real-life situations mathematically, interpret inequalities, and apply systematic methods to find optimal results.
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it can be nice when you help me to pass
Hi Matovu! Thank you for your comment! Wishing you best of luck with your studies!
Solve the following LP problem using the Graphical Method
1. A steel producer makes two types of steel: regular steel and special steel. A ton of regular steel requires 2 hours in the open-hearth furnace and 5 hours in the soaking pit; a ton of special steel requires 2 hours in the open-hearth furnace and 3 hours in the soaking pit. The open-hearth furnace is available 8 hours per day and the soaking pit is available 15 hours per day. The profit on a ton of regular steel is P4,200 and it is P5,000 on a ton of special steel. Determine how many tons of each type of steel should be made to maximize . Please answer this
Hi there! 😊
Thanks for your comment. That’s a great linear programming problem, and it’s perfect for practicing the Graphical Method. While we can’t provide a full solution here in the comments, you can find similar step-by-step problems and solutions on our resources site that cover real-world LP scenarios like this one. They’ll guide you through setting up constraints, plotting the feasible region, and finding the optimal solution.
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Thank you for your comment! 😊
If you’re looking for more scenario-based linear programming questions to practice, we have a collection of word problems and detailed solutions available on our Superprof resources site. These examples are designed to help you apply linear programming concepts to real-world situations.