SACE Stage 2 General Mathematics cheat sheet

Simple interest: I = P·r·t, where r is the annual rate as a decimal and t is time in years; total amount A = P(1 + rt).

Compound interest: A = P(1 + r)^n, where n is the number of compounding periods and r is the rate per period (annual rate divided by compounds per year).

Effective annual rate from nominal rate r compounded m times per year: EAR = (1 + r/m)^m − 1; compare rates only after converting to EAR.

Appreciation and depreciation use the same recurrence A = P(1 ± r)^n; reducing-balance depreciation uses (1 − r), flat-rate depreciation subtracts a fixed amount each period.

Recurrence form for compound growth: V(n+1) = V(n)·(1 + r) + d, where d is a regular addition (positive for deposits, negative for withdrawals).

Annuity future value: FV = M·[((1 + r)^n − 1)/r], where M is the regular payment and r the rate per compounding period.

Annuity present value (loan amount): PV = M·[(1 − (1 + r)^(−n))/r]; solve for M to find the repayment on a reducing-balance loan.

Inflation and purchasing power: real value = nominal value ÷ (1 + i)^n, where i is the inflation rate per period.

Total cost of a loan = number of repayments × repayment amount; interest paid = total cost − principal borrowed.

Stages of the process: define the problem, collect data, organise and display, summarise with statistics, interpret, and communicate findings with appropriate caveats.

Categorical variables are summarised with counts and percentages; numerical variables with mean, median, range, IQR and standard deviation.

Five-number summary: minimum, Q1, median, Q3, maximum; IQR = Q3 − Q1; outliers lie below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures spread about the mean; roughly 68% of roughly symmetric data lie within one standard deviation and 95% within two.

Pearson correlation r lies between −1 and 1 and measures only the strength of a linear association; r is unaffected by which variable is explanatory.

Least-squares regression line: y = a + bx with b = r·(s_y/s_x) and a = ȳ − b·x̄; the line passes through the point (x̄, ȳ).

Coefficient of determination r² gives the proportion of variation in the response variable explained by the linear model.

Causation cannot be inferred from an observed association; confounding variables and the sampling method must be considered when interpreting results.

A random sample supports generalisation to the population; a biased or self-selected sample does not.

Linear model y = mx + c: constant gradient m = (y₂ − y₁)/(x₂ − x₁); equal first differences in a table signal a linear relationship.

Simultaneous equations give the break-even point: set cost C(x) = revenue R(x) and solve for x, then substitute to find the dollar value.

Profit = revenue − cost; break-even occurs where profit equals zero, with profit positive when revenue exceeds cost.

Quadratic model y = ax² + bx + c: equal second differences signal a quadratic; the vertex x = −b/(2a) gives the maximum or minimum.

Turning points and intercepts of a quadratic are found by completing the square or using x = −b/(2a); the axis of symmetry is x = −b/(2a).

Exponential model y = A·b^x: a constant ratio between successive y-values signals exponential change; b > 1 growth, 0 < b < 1 decay.

Power model y = A·x^n becomes linear under logarithms: log y = log A + n·log x, so a log–log plot has gradient n.

Exponential model linearised: log y = log A + x·log b, so a semi-log plot has gradient log b and intercept log A.

Inverse variation y = k/x gives a decreasing curve with a horizontal asymptote at y = 0; k is found from one data pair.

A network has vertices (nodes) and edges (arcs); a directed edge has an arrow, a weighted edge carries a number such as distance or cost.

The degree of a vertex is the number of edges meeting it; in a directed network, indegree counts incoming and outdegree counts outgoing edges.

Handshake lemma: the sum of all vertex degrees equals twice the number of edges, so the number of odd-degree vertices is even.

A connected network has a path between every pair of vertices; an Eulerian circuit exists only if every vertex has even degree and the network is connected.

An Eulerian trail (open) exists when exactly two vertices have odd degree; those two vertices are the start and end.

A tree is a connected network with no cycles; a spanning tree on n vertices has exactly n − 1 edges.

Minimum spanning tree: apply Kruskal's algorithm by repeatedly adding the cheapest edge that does not create a cycle.

Shortest path: apply Dijkstra's algorithm, repeatedly fixing the vertex with the smallest known distance and updating its neighbours.

Matrix multiplication is defined only when the number of columns of the first equals the number of rows of the second; entry (i, j) is the row-i by column-j dot product.

An adjacency matrix A has aᵢⱼ = 1 when an edge runs from vertex i to vertex j; the entry in A^n counts walks of length n between vertices.

A recurrence relation defines each term from earlier terms, e.g. tₙ₊₁ = k·tₙ + c, together with an initial value t₁.

First-order linear recurrence tₙ₊₁ = k·tₙ + c has a fixed point at t = c/(1 − k) when k ≠ 1; the sequence converges only if |k| < 1.

Arithmetic sequences have a constant first difference d: tₙ = t₁ + (n − 1)d, and the sum of n terms is Sₙ = (n/2)(2t₁ + (n − 1)d).

Geometric sequences have a constant ratio r: tₙ = t₁·r^(n−1), and the sum of n terms is Sₙ = t₁(r^n − 1)/(r − 1) for r ≠ 1.

A sequence converges to a limit L if terms approach L as n increases; for tₙ₊₁ = k·tₙ + c with |k| < 1, L = c/(1 − k).

Transition (Leslie-style) matrices model change between states: state(n+1) = T × state(n), where T holds the transition proportions.

In a transition matrix each column sums to 1 when it records proportions moving into the next state; the steady state satisfies T·s = s.

Graphs of recurrence-generated sequences show discrete points, not a continuous curve, so interpolation between terms is not valid.

Simple interest: I = P·r·t, where r is the annual rate as a decimal and t is time in years; total amount A = P(1 + rt).

Compound interest: A = P(1 + r)^n, where n is the number of compounding periods and r is the rate per period (annual rate divided by compounds per year).

Effective annual rate from nominal rate r compounded m times per year: EAR = (1 + r/m)^m − 1; compare rates only after converting to EAR.

Appreciation and depreciation use the same recurrence A = P(1 ± r)^n; reducing-balance depreciation uses (1 − r), flat-rate depreciation subtracts a fixed amount each period.

Recurrence form for compound growth: V(n+1) = V(n)·(1 + r) + d, where d is a regular addition (positive for deposits, negative for withdrawals).

Annuity future value: FV = M·[((1 + r)^n − 1)/r], where M is the regular payment and r the rate per compounding period.

Annuity present value (loan amount): PV = M·[(1 − (1 + r)^(−n))/r]; solve for M to find the repayment on a reducing-balance loan.

Inflation and purchasing power: real value = nominal value ÷ (1 + i)^n, where i is the inflation rate per period.

Total cost of a loan = number of repayments × repayment amount; interest paid = total cost − principal borrowed.

Stages of the process: define the problem, collect data, organise and display, summarise with statistics, interpret, and communicate findings with appropriate caveats.

Categorical variables are summarised with counts and percentages; numerical variables with mean, median, range, IQR and standard deviation.

Five-number summary: minimum, Q1, median, Q3, maximum; IQR = Q3 − Q1; outliers lie below Q1 − 1.5·IQR or above Q3 + 1.5·IQR.

Standard deviation measures spread about the mean; roughly 68% of roughly symmetric data lie within one standard deviation and 95% within two.

Pearson correlation r lies between −1 and 1 and measures only the strength of a linear association; r is unaffected by which variable is explanatory.

Least-squares regression line: y = a + bx with b = r·(s_y/s_x) and a = ȳ − b·x̄; the line passes through the point (x̄, ȳ).

Coefficient of determination r² gives the proportion of variation in the response variable explained by the linear model.

Causation cannot be inferred from an observed association; confounding variables and the sampling method must be considered when interpreting results.

A random sample supports generalisation to the population; a biased or self-selected sample does not.

Linear model y = mx + c: constant gradient m = (y₂ − y₁)/(x₂ − x₁); equal first differences in a table signal a linear relationship.

Simultaneous equations give the break-even point: set cost C(x) = revenue R(x) and solve for x, then substitute to find the dollar value.

Profit = revenue − cost; break-even occurs where profit equals zero, with profit positive when revenue exceeds cost.

Quadratic model y = ax² + bx + c: equal second differences signal a quadratic; the vertex x = −b/(2a) gives the maximum or minimum.

Turning points and intercepts of a quadratic are found by completing the square or using x = −b/(2a); the axis of symmetry is x = −b/(2a).

Exponential model y = A·b^x: a constant ratio between successive y-values signals exponential change; b > 1 growth, 0 < b < 1 decay.

Power model y = A·x^n becomes linear under logarithms: log y = log A + n·log x, so a log–log plot has gradient n.

Exponential model linearised: log y = log A + x·log b, so a semi-log plot has gradient log b and intercept log A.

Inverse variation y = k/x gives a decreasing curve with a horizontal asymptote at y = 0; k is found from one data pair.

A network has vertices (nodes) and edges (arcs); a directed edge has an arrow, a weighted edge carries a number such as distance or cost.

The degree of a vertex is the number of edges meeting it; in a directed network, indegree counts incoming and outdegree counts outgoing edges.

Handshake lemma: the sum of all vertex degrees equals twice the number of edges, so the number of odd-degree vertices is even.

A connected network has a path between every pair of vertices; an Eulerian circuit exists only if every vertex has even degree and the network is connected.

An Eulerian trail (open) exists when exactly two vertices have odd degree; those two vertices are the start and end.

A tree is a connected network with no cycles; a spanning tree on n vertices has exactly n − 1 edges.

Minimum spanning tree: apply Kruskal's algorithm by repeatedly adding the cheapest edge that does not create a cycle.

Shortest path: apply Dijkstra's algorithm, repeatedly fixing the vertex with the smallest known distance and updating its neighbours.

Matrix multiplication is defined only when the number of columns of the first equals the number of rows of the second; entry (i, j) is the row-i by column-j dot product.

An adjacency matrix A has aᵢⱼ = 1 when an edge runs from vertex i to vertex j; the entry in A^n counts walks of length n between vertices.

A recurrence relation defines each term from earlier terms, e.g. tₙ₊₁ = k·tₙ + c, together with an initial value t₁.

First-order linear recurrence tₙ₊₁ = k·tₙ + c has a fixed point at t = c/(1 − k) when k ≠ 1; the sequence converges only if |k| < 1.

Arithmetic sequences have a constant first difference d: tₙ = t₁ + (n − 1)d, and the sum of n terms is Sₙ = (n/2)(2t₁ + (n − 1)d).

Geometric sequences have a constant ratio r: tₙ = t₁·r^(n−1), and the sum of n terms is Sₙ = t₁(r^n − 1)/(r − 1) for r ≠ 1.

A sequence converges to a limit L if terms approach L as n increases; for tₙ₊₁ = k·tₙ + c with |k| < 1, L = c/(1 − k).

Transition (Leslie-style) matrices model change between states: state(n+1) = T × state(n), where T holds the transition proportions.

In a transition matrix each column sums to 1 when it records proportions moving into the next state; the steady state satisfies T·s = s.

Graphs of recurrence-generated sequences show discrete points, not a continuous curve, so interpolation between terms is not valid.