Formula Sheet
This sheet lists only the applied formulas you will invoke directly when solving exam problems — approximately one to four per module. Intermediate algebraic properties (fraction rules, power laws, notable identities as stand-alone definitions) are covered in each module’s Core Concepts section and in the glossary.
Module 1 — Numerical Calculations
Percentage change
\[ \Delta\% = \frac{V_{\text{final}} - V_{\text{initial}}} {V_{\text{initial}}} \times 100 \]
Measures the relative change from an initial to a final value. The initial value is always the reference base.
Index number
\[ I = \frac{V_{\text{current}}}{V_{\text{reference}}} \times 100 \]
An index above 100 signals an increase relative to the reference period; below 100 signals a decrease.
Rule of proportionality (rule of three)
\[ \frac{a}{b} = \frac{c}{x} \;\Longrightarrow\; x = \frac{b \times c}{a} \]
Used to scale quantities proportionally when the ratio is known.
Combined factor for two successive changes
\[ (1 + t_1)(1 + t_2) \]
Two successive percentage changes \(t_1\) and \(t_2\) (expressed as decimals) multiply rather than add. For discounts, replace \((1+t)\) by \((1-t)\).
Module 2 — Algebra and Equations
Distributive law (expanding)
\[ a(b + c) = ab + ac \]
Every term inside the parentheses must be multiplied by the factor outside.
Break-even quantity
\[ Q^* = \frac{F}{p - v} \]
\(F\) = fixed cost, \(p\) = selling price per unit, \(v\) = variable cost per unit. Requires \(p > v\) (positive contribution margin).
Module 3 — Graphs, Systems, and Inequalities
Slope of a line through two points
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
A positive slope indicates an increasing line; negative indicates decreasing; zero indicates horizontal.
Affine function (slope-intercept form)
\[ y = mx + p \]
\(m\) is the slope (rate of change); \(p\) is the \(y\)-intercept (value when \(x = 0\)).
Intersection of two affine functions
\[ x^* = \frac{p_2 - p_1}{m_1 - m_2}, \qquad y^* = m_1\,x^* + p_1 \]
Valid when \(m_1 \neq m_2\) (non-parallel lines). Models the equilibrium or crossover point between two linear relationships.
Module 4 — Numerical Sequences
Arithmetic sequence — general term
\[ u_n = u_0 + n\,d \]
\(u_0\) is the first term; \(d\) is the common difference (constant addition per step).
Arithmetic sequence — sum of the first \(n+1\) terms
\[ S_n = \frac{(n+1)(u_0 + u_n)}{2} = (\text{number of terms}) \times \frac{\text{first} + \text{last}}{2} \]
Geometric sequence — general term
\[ u_n = u_0 \times r^{\,n} \]
\(u_0\) is the first term; \(r\) is the common ratio (constant factor per step).
Geometric sequence — sum of the first \(n+1\) terms (\(r \neq 1\))
\[ S_n = u_0 \,\frac{1 - r^{\,n+1}}{1 - r} \]
Module 5 — Financial Mathematics
Simple interest — future value
\[ V_s = P\,(1 + i\,n) \]
\(P\) = principal, \(i\) = interest rate per period, \(n\) = number of periods. Interest is computed only on the original principal each period.
Compound interest — future value
\[ V_c = P\!\left(1 + \frac{i}{k}\right)^{nk} \]
\(k\) = capitalisation frequency (e.g., \(k=12\) for monthly). Interest earns interest; the gap over simple interest grows with time.
Present value (annual compounding)
\[ P = \frac{V_c}{(1 + i)^n} \]
The initial capital that produces a future value \(V_c\) at rate \(i\) after \(n\) years. Used to compare cash flows at different points in time.
Module 6 — Functions
Discriminant of a quadratic
\[ \Delta = b^2 - 4ac \]
For \(ax^2 + bx + c = 0\): if \(\Delta > 0\) two real roots; \(\Delta = 0\) one double root; \(\Delta < 0\) no real root.
Quadratic formula
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
The roots of \(ax^2 + bx + c = 0\) when \(\Delta \geq 0\).
Inverse relationship between \(\ln\) and \(\exp\)
\[ \ln(e^x) = x, \qquad e^{\ln x} = x \quad (x > 0) \]
Used to solve equations with exponential or logarithmic unknowns (e.g., finding the time to double a capital).
Module 7 — Derivatives and Optimisation
Derivative of a power function
\[ (x^n)' = n\,x^{n-1} \]
Applies to any real exponent \(n\). Special cases: \((x^2)' = 2x\), \((x)' = 1\), \((c)' = 0\) for constants.
Linearity of the derivative
\[ (af + bg)' = a f' + b g' \]
The derivative of a linear combination is the same linear combination of the derivatives.
Necessary condition for a local extremum
\[ f'(x^*) = 0 \]
At a maximum or minimum of a differentiable function the tangent is horizontal. For \(f(x) = ax^2 + bx + c\): maximum if \(a < 0\), minimum if \(a > 0\).