Variable
A quantity that can change in a model.
Use when
Use variables to describe inputs and outputs clearly.
Key method
Choose a letter and define it with units.
Example
Let d be distance travelled in miles.
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Higher Applications of Mathematics
Mathematical Modelling reference
Quick reference cards for variables, formulae, model types, units, error, tolerance, assumptions and limitations.
Formula
A rule connecting variables.
Use when
Use when the relationship between quantities is known.
Key method
Substitute known values, then calculate the unknown value.
Example
cost = 0.18 × units used.
Model
A simplified mathematical description of a real situation.
Use when
Use to estimate, predict or compare decisions.
Key method
State assumptions, calculate outputs and check limitations.
Example
A fuel model may assume constant miles per litre.
Independent variable
The input variable that is changed or chosen.
Use when
Use when deciding what the model depends on.
Key method
Place it on the x-axis for graphs.
Example
Number of months is independent in a savings model.
Dependent variable
The output variable affected by the input.
Use when
Use when interpreting what the model predicts.
Key method
Place it on the y-axis for graphs.
Example
Total cost depends on the number of units used.
Linear model
A model with a constant rate of change.
Use when
Use when the output changes by the same amount each step.
Key method
y = mx + c.
Example
A taxi fare with fixed charge plus cost per mile is linear.
Gradient
The rate of change in a linear model.
Use when
Use to interpret cost per unit, speed or change per step.
Key method
gradient = change in y / change in x.
Example
A gradient of 0.22 means 22p per unit.
Intercept
The output when the input is zero.
Use when
Use to interpret starting values or fixed charges.
Key method
In y = mx + c, c is the intercept.
Example
A 3.50 intercept may be a booking fee.
Quadratic model
A curved model involving a squared term.
Use when
Use when the change itself changes, such as area or projectile-style contexts.
Key method
Use a formula such as y = ax² + bx + c.
Example
A garden area model can be quadratic if both dimensions change.
Exponential model
A model where values multiply by a constant factor.
Use when
Use for repeated percentage growth or decay.
Key method
final value = starting value × factorⁿ.
Example
A car losing 15% per year has decay factor 0.85.
Growth factor
The multiplier for repeated percentage increase.
Use when
Use with exponential growth.
Key method
growth factor = 1 + rate.
Example
4% growth has factor 1.04.
Decay factor
The multiplier for repeated percentage decrease.
Use when
Use with depreciation or reducing quantities.
Key method
decay factor = 1 - rate.
Example
12% decrease has factor 0.88.
Tolerance
The allowed range around a target measurement.
Use when
Use for manufacturing, building and measurement checks.
Key method
lower limit = target - tolerance; upper limit = target + tolerance.
Example
40 mm ± 0.5 mm allows 39.5 mm to 40.5 mm.
Absolute error
The size of an error in original units.
Use when
Use when comparing a measured value with a true or target value.
Key method
absolute error = |measured - actual|.
Example
82.4 cm instead of 82.0 cm gives 0.4 cm error.
Relative/percentage error
Error compared with the size of the value.
Use when
Use when comparing errors across different scales.
Key method
percentage error = absolute error ÷ actual value × 100%.
Example
2 g error on 50 g is 4%.
Checking units
Making sure calculations and answers use sensible units.
Use when
Use before trusting a model output.
Key method
Track units through the calculation and convert where needed.
Example
Minutes must be converted to hours before using mph.
Model assumptions and limitations
The simplifications and weaknesses in a model.
Use when
Use when judging whether a prediction is reliable.
Key method
State what has been assumed and what might affect accuracy.
Example
A travel model may ignore traffic and weather.