General Mathematics Unit 4 Subject Matter

General Mathematics Unit 4: Investing and Networking

GM Unit 4 Topic 1: Loans, Investments and Annuities 1

Sub-topic: Compound Interest Loans and Investments (6 hours)

• Use a recurrence relation to model a compound interest loan or investment.

• where is total amount at the beginning of the period, is total amount at the beginning of the period, and where is interest rate per compounding period

• Use the compound interest formula to model a compound interest loan or investment.

• where is total amount, is principal, is interest rate per compounding period and is number of compounding periods

• Calculate the effective annual rate of interest, , 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.

• where is interest rate per compounding period and is number of compounding periods per year

• Solve practical problems involving compound interest loans or investments, including determining the total amount of the loan or investment, total interest, principal, interest rate per year and per compounding period, and the effect of the interest rate and number of compounding periods on the total amount.

Sub-topic: Present Value of Ordinary Annuities (6 hours)

• Use a recurrence relation to model the present value of an ordinary annuity, e.g. reducing balance loan or retirement pension with periodic payments where interest is calculated before the periodic payment is made.

• where  is total amount at the beginning of the period,
  is total amount at the beginning of the period, is periodic payment, and where is interest rate per compounding period

• Use the present value annuity formula to model the present value of an ordinary annuity, e.g. reducing balance loan or retirement pension with periodic payments where interest is calculated before the periodic payment is made.

• where is total amount, is periodic payment,  is interest rate per compounding period and is number of compounding periods

• Solve practical problems involving the present value of an ordinary annuity, including determining the total amount of the annuity, periodic payment, total payments and total interest.

GM Unit 4 Topic 2: Loans, Investments and Annuities 2

Sub-topic: Perpetuities and Future Value of Ordinary Annuities (8 hours)

• Use a recurrence relation to model the future value of an ordinary annuity, e.g. compound interest investment with periodic payments where interest is calculated before the periodic payment is made.

• where is total amount at the beginning of the period, is total amount at the beginning of the period, is periodic payment and where  is interest rate per compounding period

• Use the future value annuity formula to model the future value of an ordinary annuity, e.g. compound interest investment with periodic payments where interest is calculated before the periodic payment is made.

• where is total amount, is periodic payment,  is interest rate per compounding period and is number of compounding periods

• Solve practical problems involving the future value of an ordinary annuity, including determining the total amount of the annuity, periodic payment, total payments and total interest.

• Use the perpetuity formula, where is total amount, is periodic payment and
   is interest rate per compounding period.

• Solve practical problems involving perpetuities, including determining the total amount of the perpetuity, periodic payment and interest rate per compounding period.

GM Unit 4 Topic 3: Graphs and Networks

Sub-topic: Graphs, Associated Terminology and the Adjacency Matrix (4 hours)

• Understand the meaning of Draw, vertex (node), edge (arc), loop, degree of a vertex, subgraph, simple graph, complete graph, bipartite graph, directed graph (digraph), weighted graph and network.

• Draw a network diagram to represent practical situations, e.g. tracks connecting camp sites in a national park, a social network, a transport network with one-way streets, the results of a round-robin sporting competition.

• Construct an adjacency matrix from a given graph or digraph.

• Construct a graph or digraph from a given adjacency matrix.

Sub-topic: Planar Graphs, Paths and Cycles (8 hours)

• Understand the meaning of planar graph and face.

• Apply Euler’s formula to solve problems relating to planar graphs.

• where is number of vertices, is number of faces and  is number of edges

• Understand the meaning of walk, trail, path, open walk, open trail, open path, closed walk, closed trail (circuit), closed path (cycle), connected graph and bridge.

• Solve practical problems to Determine the shortest path between two vertices in a weighted graph (by trial-and-error methods only).

• Understand the meaning of Eulerian trail, semi-Eulerian graph, Eulerian circuit and Eulerian graph, and the conditions for their existence.

• Solve practical problems involving semi-Eulerian graphs and Eulerian graphs.

• Understand the meaning of Hamiltonian path, semi-Hamiltonian graph, Hamiltonian cycle and Hamiltonian graph.

• Solve practical problems involving semi-Hamiltonian graphs and Hamiltonian graphs (by trial-and-error methods only).

GM Unit 4 Topic 4: Networks and Decision Mathematics 1

Sub-topic: Trees and Minimum Connector Problems (4 hours)

• Understand the meaning of tree, spanning tree and minimum spanning tree.

• Determine a minimum spanning tree in a weighted connected graph.

• Solve practical problems involving minimum spanning trees, e.g. minimising the length of cable needed to provide power from a single power station to substations in several towns.

Sub-topic: Project Planning and Scheduling Using Critical Path Analysis (CPA) (8 hours)

• Construct a project network diagram (activity on arc) to represent the durations and interdependencies of activities that must be completed during the project (excluding dummy activities).

• Use forward and backward scanning to determine the earliest starting time (EST) and latest starting time (LST) for each activity in the project.

• Use ESTs and LSTs to locate the critical path/s for a project.

• Use the critical path to determine the minimum time for a project to be completed.

• Calculate float times for non-critical activities.

• Solve small-scale practical problems involving critical path analysis.

GM Unit 4 Topic 5: Networks and Decision Mathematics 2

Sub-topic: Flow Networks (4 hours)

• Understand the meaning of 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.

• Solve small-scale practical problems involving flow networks (up to 8 possible cuts), including determining the minimum cut and the maximum flow.

Sub-topic: Assigning Order and the Hungarian Algorithm (7 hours)

• 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 for small-scale practical problems by inspection.

• Use the Hungarian algorithm ( up to square matrices) to determine the optimum (minimum and maximum) assignment/s for larger practical problems.