Achievement Standard Mathematics and Statistics 91574:
Apply linear programming methods in solving problems
Resource reference: Mathematics and Statistics 3.2B
Resource title: Mathematics Test
Credits: 3
Achievement
Apply linear programming methods
in solving problems.
Achievement with Merit
Achievement with Excellence
Apply linear programming methods,
using relational thinking, in solving
problems.
Apply linear programming methods,
using extended abstract thinking, in
solving problems.
Introduction
This activity requires you to use linear programming to investigate a situation that involves maximising a function in
two variables subject to various constraints. You will present your findings as a written report, supported by graphs,
equations and relevant calculations. The quality of your reasoning and how well you link this to the context will
determine the overall grade.
Context
Yiping is studying for a mathematics test in which there are two sections: multi-choice and short answer.
Each question in section A is multi-choice and scores 3 marks.
Each question in section B is short-answer and scores 6 marks.
It is two hours before the test and Yiping has not studied at all; in fact, he knows nothing! His teacher has given him
very similar questions to practise on and fortunately Yiping has an excellent short-term memory.
Questions in section A take Yiping 5 minutes to memorise and questions in section B take him 8 minutes to memorise.
Yiping has not been told exactly how many questions will be in each section, however his teacher has told him to
expect at least three questions in section A for every 2 questions in section B.
Task
This activity requires you to:

Use linear programming to model the constraints Yiping needs to consider when studying for his test
AND

Find out how many questions from each section Yiping should memorise in order to maximise his test mark.
You will present your findings as a written report, supported by graphs, equations and relevant calculations. As you
write your report take care to clearly communicate your findings using appropriate mathematical statements. Include
graphs, equations, and relevant calculations.
SCENARIO ONE
Yiping has two hours to study for the test. How many questions should Yiping memorise from each section in order to
maximise his test mark? State the maximum possible test mark.
SCENARIO TWO
Yiping’s friend Kailash has already sat the test. Kailash tells Yiping there are a total of 25 questions in the test.
Yiping drinks an Orange Cow energy drink that speeds up his ability to memorise questions in section A.
At what rate does Yiping need to memorise the questions in section A so that he has multiple ways of maximising his
test mark? State the number of questions that he should memorise in each section and the test mark.
Assessment schedule: Mathematics and Statistics 91574 Pizza Shack
Evidence/Judgements for Achievement
Evidence/Judgements for Achievement with Merit
The student has applied linear programming methods
in solving problems to make a recommendation about
the optimum number of birds.
The student has applied linear programming methods,
using relational thinking, in solving problems to make
a recommendation about the optimum number of
birds.
The student has linear programming methods, using
extended abstract thinking, in solving problems to
make a recommendation about the optimum number
of birds and the effect on profit.
The student has demonstrated relational thinking by
 forming and using a model AND
 relating their findings to the context, or
communicating their thinking using appropriate
mathematical statements
The student has demonstrated extended abstract
thinking by
 devising a strategy to investigate or solve a
problem AND
 using correct mathematical statements or
communicating mathematical insight.
The student has demonstrated this by



selecting and using methods
demonstrating knowledge of concepts and terms
communicating using appropriate
representations.
For example:
 Finding the equation of a linear inequality, for
example,
Time: 5x + 8y < 120
Ratio: 2x ≥ 3y
Non negativity: x ≥ 0, y ≥ 0

Representing the feasible region graphically,
correctly showing at least two of the constraints.
For example:
The student correctly graphs the linear system and
identifies the feasible region. They have
recommended the optimal number pizzas maximise
the profit.
Evidence/Judgements for Achievement with
Excellence
For example:
The student has formed and used a system of linear
constraints to find the optimal solution.
Scenario (1)
Scenario (1)
Gives all integer values and maximum mark:
See graph below
Gives one integer value:
(12, 7), (14, 6), (16, 5) max 78 marks
Give potential vertices that maximises profit for the
feasible region OR gradient method
(12, 7) 78
OR
(14, 6) 78
Scenario (2)

Objective Function:
New constraint:

M = 3x + 6y
(16, 5) 78
x + y < 25

Vertices: (11.6, 7.74) 81
Answer in context and explained or calculations or
objective function gradient used.

(24, 0) 72
Multiple solutions giving maximum of 78 Marks

If constraint has the same gradient as the objective
function i.e. -1/2
5x + 8y < 120
kx + 8y < 120 Gradient must be -1/2 and hence k = 4
Solutions: (14, 8), (16, 7), (18, 6), (20, 5) Max 90
marks
Final grades will be decided using professional judgement based on a holistic examination of the evidence provided against the criteria in the
Achievement Standard.