These seven original questions follow one path: write a rule, state its assumptions, test it against data, and judge where it stops working. Try each one on paper before opening the answer.
Work in order. The early questions build the rule, and the later ones ask you to judge it.
Questions
Question 1. A taxi charges RM4 to start and RM1.50 for every kilometre. Write a rule for the fare C in terms of the distance d, then find the fare for 12 km.
Answer
The fare has a fixed part and a part that grows with distance, so C = 4 + 1.5d.
For d = 12: C = 4 + 1.5(12) = 4 + 18 = RM22.
Assumption: the rate stays the same for the whole trip, with no waiting charge or night surcharge.
Question 2. A tank holds 600 litres. A tap fills it at 12 litres per minute. Write a model for the time to fill the tank and state two assumptions.
Answer
Time = 600 ÷ 12 = 50 minutes.
Assumptions: the tank starts empty, and the tap flows at a constant 12 litres per minute with no leaks.
If the tank already held 150 litres, the time would be (600 − 150) ÷ 12 = 37.5 minutes, so the first assumption matters.
Question 3. A student proposes y = 3x + 2. The measured values are (1, 5.1), (2, 8.0), (3, 10.9) and (4, 14.2). Does the model fit?
Answer
Model values: 5, 8, 11, 14.
Differences (measured − model): 0.1, 0, −0.1, 0.2.
Every difference is 0.2 or less, and the gaps do not keep growing in one direction. The model fits well for x from 1 to 4.
Question 4. Plan A has a fixed charge of RM30 plus RM0.10 for each call minute. Plan B has a fixed charge of RM50 and unlimited calls. Find when the two plans cost the same, then advise a user with 150 call minutes.
Answer
Plan A: 30 + 0.10m. Plan B: 50.
Set 30 + 0.10m = 50, so 0.10m = 20 and m = 200.
At 150 minutes, Plan A costs 30 + 15 = RM45, which is less than RM50. Plan A is cheaper below 200 call minutes.
Question 5. A plant grows by h = 5 + 2t centimetres after t weeks. It stops growing at 40 cm. After how many weeks does the model stop being valid, and what should the model say after that?
Answer
Solve 5 + 2t = 40, so 2t = 35 and t = 17.5.
The linear model is valid only for 0 ≤ t ≤ 17.5. After that, the refined model is h = 40.
A linear rule keeps rising forever, which is why the original model predicts 65 cm at week 30.
Question 6. The data are (1, 3), (2, 7), (3, 13) and (4, 21). Model A is y = 4x − 1 and Model B is y = x² + x + 1. Which model is better?
Answer
Model A gives 3, 7, 11, 15. The differences from the data are 0, 0, 2 and 6, and they grow.
Model B gives 3, 7, 13, 21, which matches every data point.
Model B is better, because the data rise faster each step and a straight line cannot follow that.
Question 7. A shop has sales of 20, 24 and 28 items on days 1, 2 and 3. Day 4 was not recorded, and day 5 shows 36. Estimate day 4 and say whether the day 5 record supports your method.
Answer
The sales rise by 4 each day, so the model is S = 16 + 4n, where n is the day number.
Day 4: 16 + 4(4) = 32 items.
Day 5 from the model: 16 + 4(5) = 36, which matches the record. The match supports the method, but it does not prove day 4 was exactly 32.
If you got these wrong
Missing the rule in questions 1 and 2 points to defining variables and assumptions and choosing a simple model for a real situation.
Questions 3 and 6 need testing a model against supplied data. Question 5 is covered in explaining limitations and refining a model.
Questions 4 and 7 use choosing a mathematical model from imperfect information. Log each slip in the mistake log and paper-error review, then build a timed set with the timed original practice session builder.
For a teacher to go through your written reasoning, see online one-to-one Mathematics tuition.