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Approximation and Errors: Paper 1 Long Questions

Ten Paper 1 questions on rounding up, down and off, the upper and lower limits of a measurement, relative and percentage error, and whether a claimed total is possible. The set opens with the rounding questions that appear in Section A(1) and ends with a difference of two measurements, where the limits have to be crossed. Detailed solutions can be requested by email.

10 questions · 37 marks · about 40 minutes · answers below each question

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

Question 1 · Modelled on 2014 P1 Q3 — a zero sits inside the number, so the 3-significant-figure answer needs a place-holding zero

(a) Round up $3074.526$ to the nearest hundred.

(b) Round down $3074.526$ to 2 decimal places.

(c) Round off $3074.526$ to 3 significant figures.

(3 marks)

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(a) $3100$ (b) $3074.52$ (c) $3070$

Round up and round down ignore the 5-or-more rule. In (c) the last zero only holds the place of the units digit.

Question 2 · Modelled on 2018 P1 Q3 — moved below 1, where the leading zeros are not significant figures

(a) Round off $0.060 495$ to 3 significant figures.

(b) Round down $0.060 495$ to 4 decimal places.

(c) Round up $0.060 495$ to 1 significant figure.

(3 marks)

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(a) $0.0605$ (b) $0.0604$ (c) $0.07$

Significant figures are counted from the 6. The zero after the 6 counts, the two zeros before it do not.

Question 3 · New sub-type, no direct past-paper model — maximum absolute error, the two limits and the percentage error from one reading

The length of a rope is measured as $18.4 m$ correct to the nearest $0.1 m$ .

(a) Write down the maximum absolute error of the measurement.

(b) Find the lower limit and the upper limit of the actual length of the rope.

(c) Find the percentage error of the measurement correct to 3 significant figures.

(3 marks)

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(a) $0.05 m$ (b) $18.35 m$ and $18.45 m$ (c) $0.272%$

The maximum absolute error is half of the unit of measurement, here half of 0.1 m.

Question 4 · Modelled on 2013 P1 Q8 — same two-step shape, with tins measured to the nearest 10 mL and a total in litres

A tin of paint is termed standard if its volume is measured as $750 mL$ correct to the nearest $10 mL$ .

(a) Find the least possible volume of a standard tin of paint.

(b) Is it possible that the total volume of 40 standard tins of paint is measured as $29.7 L$ correct to the nearest $0.1 L$ ? Explain your answer.

(4 marks)

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(a) $745 mL$ (b) Not possible. The least total is $29.8 L$ , but a reading of $29.7 L$ needs a total below $29.75 L$ .

Compare the least possible total with the upper limit of the reading 29.7 L, not with 29.7 itself.

Question 5 · Modelled on 2014 P2 Q11 — the multiple-choice idea written out in full, with the perimeter asked first

The length and the width of a rectangular photo are measured as $12 cm$ and $7 cm$ respectively, correct to the nearest cm.

(a) Find the least possible perimeter of the photo.

(b) Let $A cm^{2}$ be the actual area of the photo. Find the range of values of $A$ .

(4 marks)

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(a) $36 cm$ (b) $74.75 \leq A < 93.75$

Both sides take their lower limits together for the least area, and their upper limits together for the greatest.

Question 6 · Modelled on 2015 P2 Q14 — the multiple-choice idea as a long question, so the working must show which limit of each measurement is used

The weight of rice in a bag is measured as $8 kg$ correct to the nearest kg. All the rice in the bag is packed into $n$ packets such that the weight of rice in each packet is measured as $150 g$ correct to the nearest $10 g$ . Find the greatest possible value of $n$ .

(3 marks)

Show answer

$58$

Most packets means the heaviest possible bag and the lightest possible packets: 8500 ÷ 145 = 58.6…, so 58.

Question 7 · New sub-type, no direct past-paper model — the relative error decides which measurement is more accurate, not the size of the error

The length of a corridor is measured as $48 m$ correct to the nearest m. The thickness of a book is measured as $2.5 cm$ correct to the nearest $0.1 cm$ .

(a) Find the relative error of each measurement.

(b) Which of the two measurements is more accurate? Explain your answer.

(4 marks)

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(a) Corridor: $\frac{1}{96}$ . Book: $\frac{1}{50}$ . (b) The corridor, since $\frac{1}{96} < \frac{1}{50}$ .

The book has the smaller absolute error (0.05 cm against 0.5 m) but the larger relative error.

Question 8 · New sub-type, no direct past-paper model — runs the percentage error backwards to recover how the measurement was made

The height of a plant is measured as $25 cm$ . The percentage error of the measurement is $2%$ .

(a) Find the maximum absolute error of the measurement.

(b) Find the upper limit of the actual height of the plant.

(c) The measurement is correct to the nearest $k cm$ . Write down the value of $k$ .

(3 marks)

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(a) $0.5 cm$ (b) $25.5 cm$ (c) $k = 1$

The maximum absolute error is half of the unit of measurement, so the unit is twice 0.5 cm.

Question 9 · Modelled on 2023 P1 Q3 — here the claim turns out to be possible, so the upper limit of the total has to be found as well

A bottle of juice is termed regular if its capacity is measured as $330 mL$ correct to the nearest $10 mL$ .

(a) Find the least possible capacity of a regular bottle.

(b) Someone claims that the total capacity of 60 regular bottles can be measured as $20.1 L$ correct to the nearest $0.1 L$ . Is the claim correct? Explain your answer.

(5 marks)

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(a) $325 mL$ (b) Correct. The total $T L$ satisfies $19.5 \leq T < 20.1$ , and any total from $20.05 L$ up to (but not including) $20.1 L$ is measured as $20.1 L$ .

The total never reaches 20.1 L, yet it can still be rounded to 20.1 L. A reading and an actual value are different things.

Question 10 · New sub-type, no direct past-paper model — a difference of two measurements, where one lower limit pairs with the other upper limit

The length of a wooden plank is measured as $240 cm$ correct to the nearest $10 cm$ . A piece is cut from the plank, and the length of the piece is measured as $85 cm$ correct to the nearest cm.

(a) Find the lower limit and the upper limit of the actual length of the plank.

(b) Let $r cm$ be the actual length of the remaining part of the plank. Find the range of values of $r$ .

(c) Someone claims that the remaining part must be longer than $1.5 m$ . Do you agree? Explain your answer.

(5 marks)

Show answer

(a) $235 cm$ and $245 cm$ (b) $149.5 < r < 160.5$ (c) Disagreed. For example, a plank of $235.2 cm$ and a piece of $85.4 cm$ leave $149.8 cm$ .

For the shortest remainder, take the shortest plank and the longest piece.

Continue practising this topic: Try the Paper 1 Challenge for harder variants, or move to Paper 2 multiple-choice — also available as Paper 2 Challenge MC.

Where this sits in the syllabus

Approximation and Errors is junior secondary content that the HKDSE keeps examining. From 2012 to 2026, Paper 2 carried a rounding question in most years and a range-of-values question in many, and Paper 1 Section A(1) has several times set a three-part rounding question or a claim about a total weight or capacity. The marks are reachable. They are usually lost by using the measured value where a limit is needed, or by pairing the wrong two limits.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

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