Every time an engineer predicts how long a bridge will last, or a doctor works out how quickly a medicine leaves the bloodstream, someone has built a mathematical model. Modelling is the art of translating a real situation into equations, solving them, and then translating the answer back into plain language. It is also the skill that examiners, employers and universities value most, because it shows you can use mathematics rather than simply recite it.
What Is a Mathematical Model?
A mathematical model is a simplified description of a real system written in the language of mathematics. The word "simplified" matters. A model of a falling stone usually ignores air resistance; a model of a town's population ignores individual births and deaths and treats the population as a smooth quantity. Good modellers decide which details matter and which can be safely left out.
The statistician George Box put it memorably: "All models are wrong, but some are useful." The aim is not perfection but a description that is accurate enough to answer the question you care about.
The Modelling Cycle
Most modelling, from school projects to professional research, follows the same cycle:
- Understand the problem. What exactly are you trying to predict or decide?
- Make assumptions. Decide what to ignore and what to keep. Write the assumptions down.
- Define variables. Give symbols to the quantities involved, with units.
- Formulate equations. Use known laws, rates of change or data patterns to link the variables.
- Solve. Use algebra, calculus, or a computer.
- Interpret. Translate the answer back into the real context.
- Validate and refine. Compare predictions with real data. If they disagree, revisit your assumptions.
Step 7 is what separates modelling from ordinary textbook exercises. A model is never finished; it is improved until it is good enough.
Common Types of Models
- Linear models describe quantities that change by a fixed amount, such as a taxi fare with a fixed charge plus a rate per kilometre.
- Exponential models describe quantities that change by a fixed percentage, such as compound interest, bacterial growth and radioactive decay.
- Differential equation models describe how a rate of change depends on the current state, such as cooling, spreading diseases and vibrating springs.
- Statistical models fit curves to data when the underlying law is unknown.
- Optimisation models find the best decision subject to constraints, such as cheapest transport routes.
Worked Example 1: Population Growth
A town has a population of 50,000 and is growing at about 3% per year. When will it reach 80,000?
Assumptions: the growth rate stays constant; migration patterns do not change suddenly.
Variables: let \( t \) be the time in years and \( P(t) \) the population.
Formulation: a 3% annual increase means multiplying by 1.03 each year, so
$$ P(t) = 50000 \times (1.03)^t $$
Solving: set \( P(t) = 80000 \).
$$ 50000 \times (1.03)^t = 80000 $$
Divide both sides by 50,000:
$$ (1.03)^t = 1.6 $$
Take natural logarithms of both sides:
$$ t \ln(1.03) = \ln(1.6) $$
$$ t = \frac{\ln(1.6)}{\ln(1.03)} = \frac{0.4700}{0.02956} \approx 15.9 $$
Interpretation: the town will reach 80,000 people in roughly 16 years. Planners can use this to schedule new schools and water supplies.
Validation: if census data after five years shows only 55,000 people instead of the predicted \( 50000 \times 1.03^5 \approx 57964 \), the growth rate assumption needs revising.
Worked Example 2: Newton's Law of Cooling
A cup of tea is poured at 90°C in a room at 25°C. After 5 minutes it has cooled to 70°C. When will it reach a comfortable 50°C?
Assumption (Newton's law of cooling): the rate of cooling is proportional to the difference between the tea's temperature and the room temperature.
Formulation: with \( T \) the temperature in °C and \( t \) in minutes,
$$ \frac{dT}{dt} = -k(T - 25) $$
Solving this differential equation gives
$$ T(t) = 25 + 65e^{-kt} $$
since the initial difference is \( 90 - 25 = 65 \).
Finding \( k \): use \( T(5) = 70 \).
$$ 70 = 25 + 65e^{-5k} $$
$$ e^{-5k} = \frac{45}{65} $$
$$ k = \frac{1}{5}\ln\left(\frac{65}{45}\right) = \frac{0.3677}{5} \approx 0.0735 \text{ per minute} $$
Answering the question: set \( T = 50 \).
$$ 50 = 25 + 65e^{-0.0735t} $$
$$ e^{-0.0735t} = \frac{25}{65} $$
$$ t = \frac{\ln(65/25)}{0.0735} = \frac{0.9555}{0.0735} \approx 13.0 \text{ minutes} $$
Interpretation: the tea reaches 50°C about 13 minutes after pouring. Notice the cooling slows down: the first 20 degrees took 5 minutes, but the next 20 took about 8 more. The model captures this naturally. Forensic scientists use exactly the same equation, run backwards, to estimate time of death.
Worked Example 3: A Linear Cost Model
A small printing business pays Rs 40,000 per month in rent and wages, and each printed book costs Rs 250 in paper and ink. Books sell for Rs 450. How many books must be sold each month to break even?
Let \( x \) be the number of books. Then
$$ \text{Cost} = 40000 + 250x, \qquad \text{Revenue} = 450x $$
Break-even means revenue equals cost:
$$ 450x = 40000 + 250x $$
$$ 200x = 40000 \quad \Rightarrow \quad x = 200 $$
The business must sell 200 books a month. Every book beyond that earns Rs 200 profit. This simple model ignores bulk discounts on paper, which a refined model could include.
Choosing Good Assumptions
The hardest part of modelling is rarely the algebra. It is choosing assumptions that are simple enough to solve yet realistic enough to trust. A few practical tips:
- Start simple. Build the easiest reasonable model first, then add complexity only if predictions are poor.
- Check units. If one side of your equation is in metres and the other in seconds, something has gone wrong.
- Test extreme cases. Does your model make sense when time is zero, or very large? An exponential population model predicts infinite people eventually, which tells you it only works over limited periods. The logistic model fixes this by adding a carrying capacity.
- State limitations honestly. A good report always says when the model should not be used.
Did You Know?
- During the COVID-19 pandemic, governments relied heavily on SIR models (Susceptible, Infected, Recovered), a system of three differential equations first proposed by Kermack and McKendrick in 1927.
- Weather forecasts are produced by solving millions of equations for air pressure, temperature and wind on a grid covering the whole planet, several times a day.
- The logistic growth model was introduced by the Belgian mathematician Pierre-François Verhulst around 1838 to describe human population growth with limited resources.
Key Takeaways
- A mathematical model is a deliberately simplified description of a real situation using equations.
- Modelling follows a cycle: understand, assume, define variables, formulate, solve, interpret and validate.
- Linear, exponential and differential equation models cover a huge range of everyday problems.
- Clearly stated assumptions and honest limitations are as important as correct algebra.
- Comparing predictions with real data is how models are tested and improved.
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