Modelling Growth and Decay
Modelling is a cycle, not a sum
Mathematical modelling is the craft of turning a real situation into mathematics, drawing an answer from it, and then checking that the answer makes sense back in the real world. It is not a single calculation but a cycle. First you formulate, deciding what matters and writing an equation for it. Then you solve, using algebra to get a result. Next you interpret, saying what that result means for the situation rather than leaving it as a bare number. Then you evaluate, asking whether the answer is sensible and fits any data you have. If it does not, you modify the model and go round again. Throughout, you report your assumptions, your method, and your findings, so that others can judge and use your work. That honesty about assumptions is what separates a model from a guess.
Choosing the right kind of model
The most important judgement in modelling growth and decay is which kind of model to use, and the choice follows the shape of the change. If a quantity changes by a fixed amount each step, the model is linear and graphs as a straight line, like wages rising by a set raise each year. If it rises and then falls, or grows at an accelerating rate tied to a square, the model is quadratic and graphs as a parabola, like the height of a thrown ball over time. If it changes by a fixed factor each step, multiplying rather than adding, the model is exponential and graphs as a steepening curve, like a population that doubles. Reading a situation for whether it adds, turns, or multiplies points you straight to the right family of model, and getting this choice right matters more than any later arithmetic.
Money grows exponentially: compound interest
Finance is where these models bite hardest, and the key idea is the difference between simple and compound interest. Simple interest pays a fixed amount each year on the original sum only, so it is a linear model: a dollar at ten percent grows by ten cents a year. Compound interest pays interest on the whole growing balance, including past interest, so each year multiplies the balance by the same factor, an exponential model: a dollar at ten percent becomes one point one to the power of the number of years. The gap looks small at first but widens steadily, and over decades it is dramatic, which is exactly why compound interest is described as the engine of long-term saving and, on a loan, of runaway debt. Modelling money well means recognising which kind of interest is at work.
Decay: growth running in reverse
Not everything grows; many quantities decay, shrinking by a fixed factor each period, and this is just exponential change pointed downward. A quantity that halves each step runs one hundred, fifty, twenty-five, twelve and a half, falling fast at first and then ever more slowly, drawing nearer to zero without ever quite arriving. The same pattern describes a radioactive substance losing half its mass each half-life, a hot drink cooling toward room temperature, or the value of a car depreciating year by year. Decay models are built the same way as growth models, with a multiplying factor, but the factor is less than one, which tips the curve downward. Recognising decay as the mirror of growth lets you reuse everything you know about exponential models.
Evaluating, modifying, and reporting
A model is only as good as its fit, so the cycle does not end at a first answer. You evaluate by comparing the model with the data or with common sense. If a straight-line model passes through the first and last points but misses the curving ones between, that mismatch is the signal that the linear choice was wrong, and you modify, perhaps to an exponential model that passes through them all. Evaluation can also flag a model that fits the data but predicts something absurd outside it, such as a population growing without limit forever, prompting a more realistic refinement. Finally you report: state the assumptions you made, the model you chose and why, the answer, and its limitations. This habit of checking, adjusting, and explaining is the real substance of modelling, and it is what makes a mathematical answer something a decision can safely rest on.