KBAT Problems
KBAT Problem: Building a Model from Scratch
A KBAT mathematical-modelling problem where you turn a messy real situation into a clean equation, solve it, then check the answer makes sense.
Model, solve, check
A phone plan charges a fixed fee plus a rate per gigabyte. Write the cost as a linear model, use it to answer the question, then check the answer against common sense, a negative or absurd result means the model or the arithmetic is wrong.
Understand the problem
An original modelling task: a stall's fixed daily cost (rent and gas) is RM120. Each packet of food costs RM1.50 to make and sells for RM3.50.
Modelling means turning this into one equation before touching numbers: decide the variable (n = packets sold), the constants (fixed cost, cost and price per packet), and what you want (profit). A good model names every quantity so the equation almost writes itself.
Plan and solve
- Profit per packet = selling price − cost = 3.50 − 1.50 = RM2.00.
- Model: profit P = 2n − 120, where n is packets sold.
- Break-even (P = 0): 2n − 120 = 0 → n = 60 packets.
- For a target profit of RM200: 2n − 120 = 200 → 2n = 320 → n = 160 packets.
Check and a variant
Check n = 160: revenue = 160 × 3.50 = RM560, cost = 120 + 160 × 1.50 = RM360, profit = RM200, it matches, and n is a sensible positive whole number. A twist: the selling price falls to RM3.00.
The model becomes P = 1.5n − 120, so a RM200 profit needs 1.5n = 320, giving n = 213.3, which rounds up to 214 packets because you cannot sell a third of a packet. Re-checking against common sense is what turns a model into an answer.
Frequently asked questions
What does 'build a model' actually mean here?
It means writing the situation as an equation before you calculate. You choose a variable for the unknown, express the fixed and per-item amounts, and combine them into a formula like P = 2n − 120.
Once the model is right, answering different questions is just substitution, which is why examiners reward the setup.
Why round 213.3 up to 214, not down?
Because 213 packets give slightly less than the RM200 target, so you need one more to reach it. When the answer must be a whole thing and you need to meet or beat a target, round up; when it is a limit you cannot exceed, round down.
Deciding which is a KBAT judgement.
How do I check a model is sensible?
Substitute your answer back and see if the numbers agree, then ask whether the result could happen in real life. Negative packets, profits larger than total sales, or impossible sizes all signal a wrong model or an arithmetic slip.
A quick reality check catches most errors before you commit to the answer.