IB Maths AI SL · Unit 1: Number and Algebra
IB Maths AI SL Approximations and Error Questions
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 the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 26 questions
- Paper 1: 21
- Paper 2: 5
- 9 easy
- 6 medium
- 4 hard
- 5 very hard
- 2 starter
- 5 worked examples
Practise Approximations and Error questions →
AI SL formula booklet
What's examined in AI SL approximations and error
The question bank covers these approximations and error question types (number of questions in brackets):
- Measurement Bounds (11)
- Rounding and Notation (8)
- Error Analysis (7)
Key formulas
- % error
- \(\varepsilon = \left|\frac{v_A - v_E}{v_E}\right| \times 100\%\)
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 $1.45\text{ m}$ and the width as $0.62\text{ m}$. 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.
1. Round the measurements to 1 s.f.: $L \approx 1\text{ m}$ and $W \approx 0.6\text{ m}$.
2. Calculate the estimated area: $A_E = 1 \times 0.6 = 0.6\text{ m}^2$.
3. Calculate the exact area: $A_A = 1.45 \times 0.62 = 0.899\text{ m}^2$.
4. Substitute into $\varepsilon = \left| \frac{v_A - v_E}{v_E} \right| \times 100\%$: $\left| \frac{0.6 - 0.899}{0.899} \right| \times 100\%$.
5. Evaluate: $33.259\ldots\%$. Percentage error $= \mathbf{33.3\%}$.
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 $14\text{ m}$ 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: $A_{min} = \mathbf{101.25\text{ m}^2}$.
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 $45\text{ km}$, 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: $19.3617\ldots \implies \mathbf{19.4\text{ km h}^{-1}}$.
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 $1 \le a < 10$ and $k \in \mathbb{Z}$.
1. Set up: $\frac{4.2 \times 10^5}{1.4 \times 10^{-3}}$.
2. Divide the coefficients: $\frac{4.2}{1.4} = 3$.
3. Subtract the exponents: $10^{5 - (-3)} = 10^{8}$.
4. Combine: $3 \times 10^8$.
5. State: $\mathbf{3 \times 10^8}$.
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 $E = \frac{1}{2}mv^2$. 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: $7.8889\ldots\% \implies \mathbf{7.89\%}$.
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.
Question 1 · easy · 5 marks · Paper 1
A student measures the length of a rectangular classroom as \(8.2 \text{ m}\). The actual, exact length of the room is \(8.5 \text{ m}\).
Calculate the percentage error in the student’s measurement.
State, with a reason, whether the student’s measurement is an overestimate or an underestimate.
Question 2 · easy · 5 marks · Paper 1
A square metal plate has a side length of \(x = 6.4 \text{ cm}\), measured correct to 1 decimal place.
Write down the lower bound and the upper bound of \(x\).
Calculate the maximum possible perimeter of the square plate.
Question 3 · 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 \(T = 4pq\).
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 \(7.5 \times 10^6\).
All 26 approximations and error questions with mark schemes →
FAQ
How many IB Maths AI SL approximations and error questions are there?
There are 26 exam-style approximations and error questions in the AI SL question bank (Paper 1: 21 · Paper 2: 5), graded 9 easy, 6 medium, 4 hard, 5 very hard, 2 starter. 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: 21 · 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 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.
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