§-Unit

WAMathematics Applications← Mathematics Applications syllabus

Unit 4

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

How do we model quantities that grow or shrink by a constant ratio?

Recognise geometric growth and decay, use recurrence relations and the explicit rule for geometric sequences, and model compound and reducing situations.

How to model constant-ratio change with geometric sequences, switch between recurrence and explicit rules, and apply them to compound interest, depreciation and population change.

8 min read →

How do regular payments build up or draw down a fund over time?

Model annuities and annuity-investments with recurrence relations and find balances, payments and the time to exhaust or reach a target.

How to model an annuity that pays out and an annuity-investment that builds up, using recurrence relations, and find balances, payment sizes and how long a fund lasts or takes to reach a target.

8 min read →

How much can flow from a source to a sink through a capacitated network?

Model flow in a directed network, find the maximum flow, and use the maximum-flow minimum-cut relationship.

How to model flow in a directed capacitated network, find the maximum flow from source to sink, identify cuts, and apply the maximum-flow minimum-cut theorem.

8 min read →

How does a loan reduce to zero when interest is charged and regular repayments are made?

Model a reducing-balance loan with a recurrence relation, build an amortisation table, and find balances, repayments and total interest.

How to model a reducing-balance loan with a recurrence relation, read and build an amortisation table, and find the balance, repayment and total interest paid.

8 min read →

How do we allocate workers to tasks for the lowest total cost?

Model an assignment problem as a bipartite graph and solve it with the Hungarian algorithm to minimise total cost.

How to model an allocation as a bipartite matching, apply the Hungarian algorithm of row and column reduction and covering zeros, and read off the optimal minimum-cost assignment.

8 min read →

How do we display data measured over time and identify the patterns within it?

Construct a time series plot and identify trend, seasonal, cyclic and irregular components.

How to construct a time series plot and recognise its four components, trend, seasonal, cyclic and irregular, as the starting point of any time series analysis.

7 min read →

How can we measure and use the relationship between two numerical variables?

Construct and interpret scatterplots, calculate the correlation coefficient and least-squares regression line, and use the line to make predictions.

How to read scatterplots, measure linear association with Pearson's r and the coefficient of determination, fit the least-squares line, and predict while judging interpolation versus extrapolation.

9 min read →

How do we schedule dependent tasks to finish a project in the shortest time?

Construct an activity network, compute earliest and latest starting times and float, and identify the critical path and minimum completion time.

How to build an activity network from a precedence table, run forward and backward passes for earliest and latest start times, compute float, and identify the critical path and minimum project duration.

8 min read →

How can a fund pay out forever without ever running down?

Model a perpetuity, find the payment that keeps the balance constant, and relate it to the interest earned each period.

How a perpetuity pays a regular amount forever by withdrawing only the interest earned, how to find the sustainable payment or required principal, and how it differs from a draw-down annuity.

7 min read →

How do we find the route of least total weight between two points in a network?

Find the shortest path between two vertices in a weighted network and interpret it in context.

How to find the shortest path between two vertices in a weighted network by systematic listing or labelling, interpret the result, and tell shortest path apart from minimum spanning tree.

7 min read →

How do we measure a seasonal effect and remove it to compare periods fairly?

Calculate seasonal indices, deseasonalise and reseasonalise a time series, and interpret seasonal indices in context.

How to calculate seasonal indices that sum to the number of seasons, deseasonalise data by dividing by the index, reseasonalise by multiplying, and interpret what each index means.

8 min read →

How do we describe a pattern over time and use it to forecast?

Plot and describe time series, smooth with moving averages, deseasonalise with seasonal indices, fit a trend line and forecast future values.

How to identify trend, seasonal and irregular components, smooth with moving averages, compute and apply seasonal indices to deseasonalise, fit a trend line and forecast.

9 min read →

How do we fit a trend line to a time series and forecast future values?

Fit a least-squares trend line to a time series, forecast future values, and reseasonalise forecasts for seasonal data.

How to fit a least-squares trend line to a time series using a numerical time variable, forecast future values, and reseasonalise a forecast when the data is seasonal.

8 min read →

How do we connect every site in a network at the least total cost?

Solve connector problems by finding a minimum spanning tree using Prim's algorithm and interpret its total weight.

How to solve a connector problem by finding a minimum spanning tree with Prim's algorithm, confirm it has n minus 1 edges, and interpret its total weight as the least connection cost.

8 min read →

How do we smooth out the noise in a time series to reveal its trend?

Smooth a time series using moving averages, including centred even-order averages, and using median smoothing.

How to smooth a time series with odd and even moving averages, apply centring for even-order averages, and use median smoothing to reveal the underlying trend.

8 min read →

How do we describe a normal population and estimate it from a sample?

Use the normal distribution and the 68-95-99.7 rule, standardise to z-scores, and construct and interpret sample proportions and confidence intervals.

How to apply the normal distribution and empirical rule, convert values to z-scores, work with sample proportions, and build and interpret confidence intervals for a population proportion.

9 min read →