§-Unit

QLDGeneral Mathematics← General Mathematics syllabus

Unit 4: Investing and networking

14 dot points, in syllabus order. Each dot point has a focused answer, with exam-style questions and worked answers where available.

Topic 1: Loans, investments and annuities 1 - how do we fairly compare two financial products with different compounding periods?

Calculate the effective annual rate of interest using i(effective) = (1 + i)^k - 1, and use the results to compare interest on loans or investments when interest is paid or charged for different compounding periods, including daily, monthly, quarterly and six-monthly

The QCE General Mathematics 2025 syllabus skill of comparing interest rates. Nominal versus effective annual rates, the formula i(effective) = (1 + i)^k - 1 with i per compounding period, daily to six-monthly compounding, why more frequent compounding raises the effective rate, and choosing the better investment or cheaper loan, with a question modelled on QCAA's style.

14 min read →

Topic 1: Loans, investments and annuities 1 - how do we model a reducing balance loan or an annuity that pays out, with a recurrence relation and the present value annuity formula?

Use the recurrence relation A(n+1) = r A(n) - d, with r = 1 + i, and the present value annuity formula A(PV) = d[(1 - (1 + i)^(-n)) / i] to model the present value of an ordinary annuity, such as a reducing balance loan or a retirement pension, and solve practical problems for the total amount, periodic payment, total payments and total interest

The QCE General Mathematics 2025 syllabus sub-topic on the present value of an ordinary annuity. Reducing balance loans and retirement pensions with the recurrence A(n+1) = rA(n) - d and the formula A(PV) = d[(1 - (1 + i)^(-n))/i], finding the loan amount, repayment, total payments, total interest and balance with a scientific calculator.

16 min read →

Topic 2: Loans, investments and annuities 2 - how do regular contributions build an invested balance, and how do we find the future value of an ordinary annuity?

Use the recurrence relation A(n+1) = r A(n) + d, with r = 1 + i, and the future value annuity formula A(FV) = d[((1 + i)^n - 1) / i] to model the future value of an ordinary annuity, such as a savings plan or superannuation with periodic contributions, and solve practical problems for the total amount, periodic payment, total payments and total interest

The QCE General Mathematics 2025 syllabus sub-topic on the future value of an ordinary annuity. Savings plans and superannuation with the recurrence A(n+1) = rA(n) + d and the formula A(FV) = d[((1 + i)^n - 1)/i], finding the future value, the regular deposit, total deposits and total interest, opening balances, lump sums and two-stage retirement problems.

16 min read →

Topic 3: Graphs and networks - how do we represent practical situations as graphs, describe them with the right terminology, and move between a graph and its adjacency matrix?

Understand the meaning of graph, vertex, edge, loop, degree of a vertex, subgraph, simple, complete, bipartite, directed and weighted graphs and networks, construct network diagrams for practical situations, and construct an adjacency matrix from a graph or digraph and a graph or digraph from an adjacency matrix

The QCE General Mathematics 2025 syllabus sub-topic on graph terminology and adjacency matrices. Vertices, edges, loops and degree, the sum of degrees, subgraphs, simple, complete, bipartite, directed and weighted graphs, drawing network diagrams for practical situations, and converting between graphs or digraphs and adjacency matrices.

16 min read →

Topic 3: Graphs and networks - how do we find the cheapest or quickest route between two points in a network?

Solve practical problems to determine the shortest path between two vertices in a weighted graph by trial-and-error methods, distinguishing the shortest path from a minimum spanning tree

The QCE General Mathematics 2025 syllabus content on shortest paths. What a shortest path is in a weighted graph, a systematic trial-and-error search that lists routes and prunes long ones, ties, required stops, road closures and new roads, converting distances and speeds to times, and how a shortest path differs from a minimum spanning tree.

15 min read →

Topic 4: Networks and decision mathematics 1 - how do we connect every vertex of a network at the least total cost?

Understand the meaning of tree, spanning tree and minimum spanning tree, determine a minimum spanning tree in a weighted connected graph, and solve practical problems involving minimum spanning trees, such as minimising the length of cable needed to connect several sites

The QCE General Mathematics 2025 syllabus sub-topic on trees and minimum connector problems. Trees, spanning trees and minimum spanning trees, Prim's algorithm from a network and from a table, Kruskal's algorithm as a check, ties, forced edges, and cost and savings problems, with a fully worked 7-vertex network.

14 min read →

Topic 4: Networks and decision mathematics 1 - how do we schedule a project and find the shortest time to complete it?

Construct a project network diagram (activity on arc, excluding dummy activities) from an activity table, use forward and backward scanning to determine the earliest starting time (EST) and latest starting time (LST) of each activity, locate the critical path, determine the minimum completion time, calculate float times for non-critical activities, and solve small-scale practical problems

The QCE General Mathematics 2025 syllabus sub-topic on project planning with critical path analysis. Building an activity-on-arc network from a precedence table, forward and backward scanning for EST and LST, the critical path and minimum completion time, float, and the effect of delays and shortened activities, with fully worked examples.

15 min read →

Topic 5: Networks and decision mathematics 2 - how much can flow from a source to a sink through a capacitated network?

Understand source node, sink node, cut, minimum cut and maximum flow, use a flow network diagram to identify a cut, determine the capacity of a cut, and solve small-scale practical problems involving flow networks (up to 8 possible cuts), including determining the minimum cut and the maximum flow

The QCE General Mathematics 2025 syllabus content on flow networks. Sources, sinks and capacities, conservation of flow, what makes a line a valid cut, calculating cut capacity when some edges point backwards, listing all cuts systematically (up to 8), the maximum-flow minimum-cut theorem, finding a flow that achieves the maximum, and upgrade questions.

16 min read →

Topic 5: Networks and decision mathematics 2 - how do we allocate workers to tasks at the least total cost?

Use a bipartite graph and its tabular or matrix form to represent possible assignments for an allocation problem, determine the optimum (minimum and maximum) assignment/s by inspection for small-scale problems, and use the Hungarian algorithm (3 x 3 up to 5 x 5 square matrices) to determine the optimum (minimum and maximum) assignment/s for larger practical problems

The QCE General Mathematics 2025 syllabus sub-topic on assigning order and the Hungarian algorithm. Bipartite graphs and cost matrices, optimum assignments by inspection, row and column reduction, covering zeros with the fewest lines, the adjustment step, maximisation, and problems with more than one optimal assignment, with fully worked 3 x 3 and 4 x 4 examples.

15 min read →

What is a planar graph, what are its faces, and how does Euler's formula v + f - e = 2 connect the numbers of vertices, faces and edges?

Understand the meaning of planar graph and face, and apply Euler's formula v + f - e = 2 to solve problems relating to planar graphs

The QCE General Mathematics 2025 syllabus content on planar graphs. What makes a graph planar, redrawing a graph to remove crossings, counting faces including the outer face, Euler's formula v + f - e = 2 for connected planar graphs, finding a missing count, combining it with the sum of degrees, and practical contexts such as circuit layouts and maps.

13 min read →

How do we describe routes through a graph as walks, trails, paths, circuits and cycles, and how do we decide whether a graph is Eulerian, semi-Eulerian, Hamiltonian or semi-Hamiltonian?

Understand walks, trails, paths, circuits, cycles, connected graphs and bridges, the conditions for Eulerian trails and circuits in semi-Eulerian and Eulerian graphs, and Hamiltonian paths and cycles in semi-Hamiltonian and Hamiltonian graphs, and solve practical problems involving them

The QCE General Mathematics 2025 syllabus route vocabulary and route problems. Walks, trails, paths, open and closed routes, circuits and cycles, connected graphs and bridges, the degree conditions for Eulerian trails and circuits, semi-Eulerian and Eulerian graphs, and Hamiltonian paths and cycles found by trial and error, with practical route problems.

16 min read →

How do we model a compound interest loan or investment with a recurrence relation and with the compound interest formula, and use them to find the amount, interest, principal, rate or time?

Use the recurrence relation A(n+1) = r A(n), with r = 1 + i, and the compound interest formula A = P(1 + i)^n to model compound interest loans and investments, and solve practical problems for the total amount, total interest, principal, interest rate and number of compounding periods

The QCE General Mathematics 2025 syllabus sub-topic on compound interest. The recurrence relation with r = 1 + i, the formula A = P(1 + i)^n, converting annual rates to rates per compounding period, finding the amount, interest, principal, rate and number of periods with a scientific calculator, and the effect of compounding frequency.

15 min read →

What is a perpetuity, and how do we use the perpetuity formula A = d/i to find the amount invested, the periodic payment or the interest rate?

Use the perpetuity formula A = d/i and solve practical problems involving perpetuities, including determining the total amount of the perpetuity, the periodic payment and the interest rate per compounding period

The QCE General Mathematics 2025 syllabus content on perpetuities. What a perpetuity is, why the balance never changes, the recurrence with the payment equal to the interest, the formula A = d/i and its rearrangements, matching the rate to the payment period, and comparing a perpetuity with an ordinary annuity that runs out.

13 min read →

Topics 4 and 5: Networks and decision mathematics 1 and 2 - how do we optimise connection, travel, flow and scheduling on a network?

Choose and apply the right network technique for a practical problem: minimum spanning trees for connection (Topic 4), critical path analysis for project scheduling (Topic 4), maximum flow and minimum cut for flow networks (Topic 5) and the Hungarian algorithm for assignment (Topic 5), distinguishing them from shortest path problems (Topic 3)

An overview of QCE General Mathematics 2025 Unit 4 Topics 4 and 5, Networks and decision mathematics 1 and 2: how to recognise whether a problem needs a minimum spanning tree, critical path analysis, a maximum flow and minimum cut, or an assignment, with one compact worked example of each and how they differ from shortest paths.

14 min read →