Mathematical Modeling and Problem Solving
Core concepts
Modeling questions ask you to translate a real situation into algebra, solve it, and interpret the answer in context. You should be able to define variables clearly, build linear and quadratic models, use proportional reasoning, and check units. Recognise common model families: constant rate (linear), acceleration or area problems (quadratic), and growth or decay (exponential).
Traps to watch
- Answering the algebra rather than the question-for example, giving x when the question asks for 2x + 5.
- Ignoring constraints: negative lengths, non-integer counts of people, or times beyond the model's range.
- Forgetting fixed costs in a linear cost model.
- Using an exponential model where growth is actually constant per unit time.
- Leaving units off the final answer.
Worked example
A taxi charges a fixed 3 plus 2 per kilometre. For a 12 km trip the cost is 3 + 2(12) = 27. If a rider has 25, solve 3 + 2k = 25, giving k = 11 km-not 12.5, which would come from dividing 25 by 2 and forgetting the fixed charge.
Rule of thumb
After solving, re-read the question, substitute your answer back into the original context, and confirm both the units and the reasonableness of the size.