A tank fills at a constant rate. At $t = 0$ min it contains $10\text{ L}$; at $t = 4$ min it contains $18\text{ L}$. Write a linear model $V(t)$.

1. Identify the points: $(0, 10)$ and $(4, 18)$. 2. Gradient : $m = \frac{18 - 10}{4 - 0} = 2$. 3. $y$-intercept : at $t = 0$, $V = 10$, so $c = 10$. 4. Formulate : $V(t) = mt + c$. 5. State : $\mathbf{V(t) = 2t + 10}$.

Machinery bought for $\text{€}20\,000$ depreciates linearly to $\text{€}8000$ over $6$ years. Calculate the exact annual rate of depreciation.

1. Identify the coordinates: $(0, 20000)$ and $(6, 8000)$. 2. Gradient : $m = \frac{8000 - 20000}{6 - 0}$. 3. Simplify : $m = \frac{-12000}{6}$. 4. Evaluate : $m = -2000$. 5. State : annual rate of depreciation $= \mathbf{\text{€}2000\text{ per year}}$.

Plumber A charges $\text{€}40 + \text{€}15h$. Plumber B charges $\text{€}60 + \text{€}10h$. Find the exact number of hours for which both plumbers charge the same total.

1. Model A : $C_A(h) = 40 + 15h$. 2. Model B : $C_B(h) = 60 + 10h$. 3. Equate : $40 + 15h = 60 + 10h$. 4. Collect : $40 + 5h = 60$. 5. Solve : $5h = 20 \implies h = \mathbf{4\text{ hours}}$.

An electrician charges a fixed callout fee of $\text{€}C$ plus an hourly rate of $\text{€}R$. A 2-hour job costs $\text{€}130$, and a 5-hour job costs $\text{€}235$. Write down two linear equations representing this information. Find the values of $C$ and $R$. Calculate the total cost of a job that takes 3.5 hours.

$C + 2R = 130$ A1 $C + 5R = 235$ A1 Subtract the first equation from the second: $3R = 105 \implies R = 35$ M1 Substitute back: $C + 70 = 130 \implies C = 60$ A1 Cost $= 60 + 3.5(35) = \text{€}182.50$ A1

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IB Maths AI SL · Unit 2: Functions

IB Maths AI SL Linear Modelling Questions

Updated October 2026

Exam-style IB Maths AI SL linear modelling questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

19 questions
Paper 1: 14
Paper 2: 5
8 easy
7 medium
1 hard
2 very hard
1 starter
3 worked examples

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🖩 How do I do this on my calculator? Keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire.

What you need to know

Slope, intercept, and interpreting the gradient in context. SL AI dresses linear models as taxi fares, phone plans, and rental costs — always ask what the gradient MEANS in the real-world context. Linear functions and modelling overview →

Bacteria doubling every hour, a virus spreading through a population, radioactive decay — SL AI's exponential models are always applied. Learn to fit y = a·b^x from two data points. Exponential growth and decay models overview →

Linear Modelling worked examples

Worked example 1: Building a linear model from two data points · easy

A tank fills at a constant rate. At $t = 0$ min it contains $10 L$ ; at $t = 4$ min it contains $18 L$ . Write a linear model $V (t)$ .

Solution

5 steps · 1 to fill in

Examiner tip: If the data gives a value at

t = 0

, the

y

-intercept is handed to you for free — no point-slope substitution needed.

Worked example 2: Rate of straight-line depreciation · medium

Machinery bought for $€ 20 000$ depreciates linearly to $€ 8000$ over $6$ years. Calculate the exact annual rate of depreciation.

Solution

5 steps · 1 to fill in

Examiner tip: "Depreciation" is the AMOUNT lost per year and is reported as a positive number, even though the gradient itself is negative.

Worked example 3: Comparing two linear cost models · hard

Plumber A charges $€ 40 + € 15 h$ . Plumber B charges $€ 60 + € 10 h$ . Find the exact number of hours for which both plumbers charge the same total.

Solution

5 steps

Examiner tip: Write both linear models

y = m x + c

explicitly BEFORE equating them — that prevents confusing fixed fees with hourly rates.

Try these IB Maths AI SL linear modelling questions

Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area. One of them has its full worked solution here, open to everyone.

Question 1 · easy · 3 marks · Paper 1

A car rental company charges a flat daily rate of $€ 40$ plus $€ 0.15$ for every kilometre driven.

Write down a linear function $C (x)$ to model the total daily cost of renting the car and driving it for $x$ kilometres.
Calculate the total cost of renting the car for one day and driving exactly $250 km$ .

Attempt it and see the mark scheme →

Question 2 · medium · 5 marks · Paper 2

An electrician charges a fixed callout fee of $€ C$ plus an hourly rate of $€ R$ . A 2-hour job costs $€ 130$ , and a 5-hour job costs $€ 235$ .

Write down two linear equations representing this information.
Find the values of $C$ and $R$ .
Calculate the total cost of a job that takes 3.5 hours.

Full worked solution Free, no sign-in

$C + 2 R = 130$ A1
$C + 5 R = 235$ A1
Subtract the first equation from the second: $3 R = 105 ⟹ R = 35$ M1
Substitute back: $C + 70 = 130 ⟹ C = 60$ A1
Cost $= 60 + 3.5 (35) = € 182.50$ A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 3 marks · Paper 1

The line $L$ passes through the points $(2, k)$ and $(6, 3)$ , and is perpendicular to the line $y = 2 x + 1$ . Find the value of $k$ .

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FAQ

How many IB Maths AI SL linear modelling questions are there?

There are 19 exam-style linear modelling questions in the AI SL question bank (Paper 1: 14 · Paper 2: 5), graded 2 very hard, 7 medium, 8 easy, 1 starter, 1 hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is linear modelling on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 14 · Paper 2: 5. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.

What else is free here

Save your answers and track your progressA free account remembers the questions you have done on every topic and brings you back to this page.

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Free ready-to-teach lesson (slides and teacher notes): Straight-line graphs (m and c).

IB Maths AI SL Linear Modelling Questions

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What you need to know

Linear Modelling worked examples

Worked example 1: Building a linear model from two data points · easy

Solution

Worked example 2: Rate of straight-line depreciation · medium

Solution

Worked example 3: Comparing two linear cost models · hard

Solution

Try these IB Maths AI SL linear modelling questions

Question 1 · easy · 3 marks · Paper 1

Question 2 · medium · 5 marks · Paper 2

Full worked solution Free, no sign-in

Question 3 · hard · 3 marks · Paper 1

Revise linear modelling for the exam

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