A rectangular garden has a length of $14\text{ m}$ and a width of $8\text{ m}$, both measured correct to the nearest metre. Calculate the minimum possible area of the garden.

1. Determine the maximum error: half of $1\text{ m}$ is $0.5\text{ m}$. 2. Find the lower bound of the length: $L_{min} = 14 - 0.5 = 13.5\text{ m}$. 3. Find the lower bound of the width: $W_{min} = 8 - 0.5 = 7.5\text{ m}$. 4. Multiply the lower bounds: $A_{min} = 13.5 \times 7.5$. 5. Evaluate : $A_{min} = \mathbf{101.25\text{ m}^2}$.

A cyclist travels a distance of $45\text{ km}$, measured to the nearest kilometre, in a time of $2.4\text{ hours}$, measured to the nearest $0.1$ of an hour. Calculate the maximum possible average speed of the cyclist.

1. State the formula: $v = \frac{d}{t}$. 2. Determine the upper bound of the distance: $d_{max} = 45.5\text{ km}$. 3. Determine the lower bound of the time: $t_{min} = 2.35\text{ hours}$. 4. Maximise the quotient: $v_{max} = \frac{45.5}{2.35}$. 5. Evaluate : $19.3617\ldots \implies \mathbf{19.4\text{ km h}^{-1}}$.

Kinetic energy is $E = \frac{1}{2}mv^2$. A student measures $m = 12\text{ kg}$ and $v = 4\text{ m/s}$, when the true values are $m = 12.4\text{ kg}$ and $v = 4.1\text{ m/s}$. Calculate the percentage error in the estimated kinetic energy.

1. Estimated energy: $E_{approx} = \frac{1}{2}(12)(4)^2 = 96\text{ J}$. 2. Exact energy: $E_{exact} = \frac{1}{2}(12.4)(4.1)^2 = 104.222\text{ J}$. 3. Recall : $\varepsilon = \left| \frac{v_A - v_E}{v_E} \right| \times 100\%$. 4. Substitute : $\left| \frac{96 - 104.222}{104.222} \right| \times 100\%$. 5. Evaluate : $7.8889\ldots\% \implies \mathbf{7.89\%}$.

Consider the values $p = 3.32 \times 10^{-5}$ and $q = 5.67 \times 10^{10}$ . Calculate the exact value of $T = 4pq$ . Write down your answer to part (a) in the form $a \times 10^k$ , where $1 \le a < 10$ and $k \in \mathbb{Z}$ . Calculate the percentage error if $T$ is approximated as $7.5 \times 10^6$ .

$T = 4(3.32 \times 10^{-5})(5.67 \times 10^{10})$ . (M1) $T = 7529760$ . A1 $7.52976 \times 10^6$ . A1A1 $PE = \left| \frac{7500000 - 7529760}{7529760} \right| \times 100\%$ . (M1) $= 0.395\%$ ( $0.395231\dots\%$ ). A1

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IB Maths AI SL · Unit 1: Number and Algebra

IB Maths AI SL Approximations and Error Questions

Q: How many IB Maths AI SL approximations and error questions are there?

There are 45 exam-style approximations and error questions in the AI SL question bank (Paper 1: 35 · Paper 2: 10), graded 16 easy, 16 medium, 8 hard, 5 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Q: Is approximations and error on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 35 · Paper 2: 10. Practise with your GDC — AI papers expect calculator methods throughout.

Q: Where can I get the mark schemes?

Open the AI SL Unit 1 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.

Updated October 2026

Exam-style IB Maths AI SL approximations and error questions with worked solutions. Start with the 5 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.

45 questions
Paper 1: 35
Paper 2: 10
16 easy
16 medium
8 hard
5 very hard
5 worked examples

Practise Approximations and Error questions →

What you need to know

Every SL AI paper hides at least one question that turns on writing an answer in a·10^k form. Getting the exponent right — and rounding to 3 significant figures — is the difference between an M-mark awarded and lost. Scientific notation and estimation overview →

What's examined in AI SL approximations and error

The question bank covers these approximations and error question types (number of questions in brackets):

Rounding and Notation (20)
Measurement Bounds (15)
Error Analysis (10)

Key formulas

% error: $ε = | \frac{v_{A} - v_{E}}{v_{E}} | \times 100 %$

In the same notation as the IB formula booklet. All AI SL formulas →

Approximations and Error worked examples

Worked example 1: Rounding and percentage error · easy

A student measures the length of a desk as $1.45 m$ and the width as $0.62 m$ . They estimate the area by rounding each measurement to 1 significant figure before multiplying. Calculate the percentage error of their estimated area compared to the exact area.

Solution

5 steps

Examiner tip: Always use the unrounded exact value in the denominator of the percentage-error formula. The absolute value bars mean your final answer must be positive.

Worked example 2: Minimum area from measurement bounds · medium

A rectangular garden has a length of $14 m$ and a width of $8 m$ , both measured correct to the nearest metre. Calculate the minimum possible area of the garden.

Solution

5 steps · 1 to fill in

Examiner tip: To minimise a product (area, volume), multiply the LOWER bounds of every dimension. Never compute the area first and then subtract

0.5

Worked example 3: Maximising a quotient with bounds · hard

A cyclist travels a distance of $45 km$ , measured to the nearest kilometre, in a time of $2.4 hours$ , measured to the nearest $0.1$ of an hour. Calculate the maximum possible average speed of the cyclist.

Solution

5 steps · 1 to fill in

Examiner tip: To maximise a quotient, make the numerator as large as possible (upper bound) and the denominator as small as possible (lower bound).

Worked example 4: Dividing in scientific notation · easy

Given $x = 4.2 \times 10^{5}$ and $y = 1.4 \times 10^{- 3}$ , calculate the exact value of $\frac{x}{y}$ in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in Z$ .

Solution

5 steps · 1 to fill in

Examiner tip: Type the calculation into your GDC to confirm, but always translate calculator output like `3E8` into proper mathematical notation (

3 \times 10^{8}

) on your exam paper.

Worked example 5: Percentage error in a kinetic-energy calculation · medium

Kinetic energy is $E = \frac{1}{2} m v^{2}$ . A student measures $m = 12 kg$ and $v = 4 m/s$ , when the true values are $m = 12.4 kg$ and $v = 4.1 m/s$ . Calculate the percentage error in the estimated kinetic energy.

Solution

5 steps · 1 to fill in

Examiner tip: Compute the final formula value for both measurements first, THEN apply the percentage-error formula. Do not compute percentage errors of the inputs and try to combine them.

Try these IB Maths AI SL approximations and error 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 · 5 marks · Paper 1

A student measures the length of a rectangular classroom as $8.2 m$ . The actual, exact length of the room is $8.5 m$ .

Calculate the percentage error in the student’s measurement.
State, with a reason, whether the student’s measurement is an overestimate or an underestimate.

Attempt it and see the mark scheme →

Question 2 · medium · 6 marks · Paper 1

Consider the values $p = 3.32 \times 10^{- 5}$ and $q = 5.67 \times 10^{10}$ .

Calculate the exact value of $T = 4 p q$ .
Write down your answer to part (a) in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in Z$ .
Calculate the percentage error if $T$ is approximated as $7.5 \times 10^{6}$ .

Full worked solution Free, no sign-in

$T = 4 (3.32 \times 10^{- 5}) (5.67 \times 10^{10})$ . (M1)
$T = 7529760$ . A1
$7.52976 \times 10^{6}$ . A1A1
$P E = | \frac{7500000 - 7529760}{7529760} | \times 100 %$ . (M1)
$= 0.395 %$ ( $0.395231 \dots %$ ). A1

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

Question 3 · hard · 4 marks · Paper 1

Write each number in the form $a \times 10^{k}$ , where $1 \leq a < 10$ and $k \in Z$ .

$0.0472 \times 10^{5}$ [2]
$560 \times 10^{- 7}$ [2]

Attempt it and see the mark scheme →

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FAQ

How many IB Maths AI SL approximations and error questions are there?

There are 45 exam-style approximations and error questions in the AI SL question bank (Paper 1: 35 · Paper 2: 10), graded 16 easy, 16 medium, 8 hard, 5 very hard. Every question has a full IB-style mark scheme (M, A and R marks).

Is approximations and error on Paper 1 or Paper 2?

Both. In the bank, Paper 1: 35 · Paper 2: 10. Practise with your GDC — AI papers expect calculator methods throughout.

Where can I get the mark schemes?

Open the AI SL Unit 1 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 lessons (slides and teacher notes): Writing numbers in standard form · Approximation & rounding · Upper & lower bounds of measurements.

IB Maths AI SL Approximations and Error Questions

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What you need to know

What's examined in AI SL approximations and error

Key formulas

Approximations and Error worked examples

Worked example 1: Rounding and percentage error · easy

Solution

Worked example 2: Minimum area from measurement bounds · medium

Solution

Worked example 3: Maximising a quotient with bounds · hard

Solution

Worked example 4: Dividing in scientific notation · easy

Solution

Worked example 5: Percentage error in a kinetic-energy calculation · medium

Solution

Try these IB Maths AI SL approximations and error questions

Question 1 · easy · 5 marks · Paper 1

Question 2 · medium · 6 marks · Paper 1

Full worked solution Free, no sign-in

Question 3 · hard · 4 marks · Paper 1

Revise approximations and error for the exam

FAQ

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