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Approximation and Errors: Paper 2 Challenge Multiple-Choice

This is the Challenge tier: ten harder multiple-choice questions where the tempting option is the right number to a different question. Each one turns on which limit of a measurement to use, or on whether an end value is included. Work the standard Paper 2 set first. Detailed solutions can be requested by email.

10 questions · about 20 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 2019 P2 Q6 — the interval straddles 0.05, so anything rounded up or down splits in two

If $0.049 96 < x < 0.050 04$ , which of the following must be true?

$x = 0.0500$ (correct to 3 significant figures)
$x = 0.05000$ (correct to 4 significant figures)
$x = 0.04$ (rounded down to 2 decimal places)
$x = 0.05$ (rounded up to 2 decimal places)

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A. A holds for every $x$ from $0.049 95$ up to $0.050 05$ , which covers the whole interval. B fails for $x = 0.049 97$ . C fails once $x \geq 0.05$ . D fails once $x > 0.05$ , where rounding up gives $0.06$ .

Question 2 · Modelled on 2020 P2 Q14 — the area is a difference, so the tempting option pairs lower with lower and upper with upper

In the figure, a rectangular hole is cut from a rectangular metal plate. All the measurements are correct to the nearest cm. Let $x cm^{2}$ be the actual area of the shaded region. Find the range of values of $x$ .

A rectangular metal plate 8 cm by 6 cm with a rectangular hole 5 cm by 4 cm cut out of it. The remaining plate is shaded.

$27.5 \leq x < 28.5$
$25.5 \leq x < 30.5$
$16.5 < x < 39.5$
$21.25 \leq x < 35.25$

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C. Smallest plate with largest hole: $7.5 \times 5.5 - 5.5 \times 4.5 = 16.5$ . Largest plate with smallest hole: $8.5 \times 6.5 - 4.5 \times 3.5 = 39.5$ . Option B subtracts lower from lower and upper from upper, the right method for the wrong operation.

Question 3 · Modelled on 2015 P2 Q14 — inverted: the number must be certain to be safe, not merely possible, so the other limit is the one that matters

A lift can carry a load of at most $680 kg$ . The weight of each box is measured as $45 kg$ correct to the nearest kg. Find the greatest number of boxes that can be put into the lift such that the total weight of the boxes is certain not to exceed $680 kg$ .

$13$
$14$
$15$
$16$

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B. To be certain, assume every box is as heavy as it can be. $14$ boxes weigh less than $14 \times 45.5 = 637 kg$ . $15$ boxes can weigh $15 \times 45.4 = 681 kg$ . Option C is $680 \div 45$ , the right number for exact weights.

Question 4 · New sub-type, no direct past-paper model — the percentage error is given and the reading is the unknown

A length is measured correct to the nearest $0.1 cm$ . If the percentage error of the measurement is $0.25%$ , then the measured length is

$2.0 cm$ .
$4.0 cm$ .
$20.0 cm$ .
$40.0 cm$ .

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C. The maximum absolute error is $0.05 cm$ , and $0.05 \div 0.25% = 20$ . Option D uses $0.1 cm$ as the error.

Question 5 · New sub-type, no direct past-paper model — statements about the rounding rules themselves

Let $x$ be a positive number. Which of the following must be true?

I. If $x$ is rounded off to 1 decimal place and the result is then rounded off to the nearest integer, the final result is the same as rounding off $x$ to the nearest integer directly.
II. The result of rounding up $x$ to the nearest integer is not less than the result of rounding off $x$ to the nearest integer.
III. The result of rounding down $x$ to 2 decimal places is not greater than $x$ .

II only
III only
I and II only
II and III only

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D. I fails for $x = 2.46$ : it becomes $2.5$ , then $3$ , but $2.46$ rounds off to $2$ . Rounding up never gives less than rounding off, and rounding down never gives more than the number itself.

Question 6 · Modelled on 2014 P2 Q11 — a cube instead of a rectangle; the tempting option adds 5% three times

The side of a solid cube is measured as $10 cm$ correct to the nearest cm. Let $V cm^{3}$ be the actual volume of the cube. Find the range of values of $V$ .

$999.5 \leq V < 1000.5$
$857.375 \leq V < 1157.625$
$850 \leq V < 1150$
$950 \leq V < 1050$

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B. $9. 5^{3} = 857.375$ and $10. 5^{3} = 1157.625$ . Option C takes $15%$ off and adds $15%$ on, which is close but not exact, and the true band is not centred on $1000$ .

Question 7 · New sub-type, no direct past-paper model — a difference of two rounded numbers, with options that differ only in whether the ends are included

It is given that $p = 8.4$ and $q = 3.7$ , each correct to 1 decimal place. Find the range of values of $p - q$ .

$4.6 < p - q < 4.8$
$4.6 \leq p - q \leq 4.8$
$4.65 \leq p - q < 4.75$
$4.5 < p - q < 4.9$

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A. $p - q > 8.35 - 3.75 = 4.6$ and $p - q < 8.45 - 3.65 = 4.8$ . Neither end is reached, because each needs an upper limit ( $q = 3.75$ or $p = 8.45$ ) that is excluded.

Question 8 · New sub-type, no direct past-paper model — a quotient, where the upper limit needs the smallest divisor

A car travels a distance which is measured as $150 km$ correct to the nearest $10 km$ . The time taken is measured as $2.0$ hours correct to the nearest $0.1$ hour. Find the upper limit of the average speed of the car, correct to 3 significant figures.

$74.4 km/h$
$75.6 km/h$
$79.5 km/h$
$77.5 km/h$

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C. Longest distance over shortest time: $155 \div 1.95 = 79.48 \dots$ Option B is $155 \div 2.05$ , which pairs the two upper limits.

Question 9 · Modelled on 2012 P2 Q14 — counts every possible value of n instead of only the greatest

The weight of each marble in a bag is measured as $5 g$ correct to the nearest g. The total weight of the $n$ marbles in the bag is measured as $1.2 kg$ correct to the nearest $0.1 kg$ . How many possible values of $n$ are there?

$21$
$48$
$67$
$68$

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D. $4.5 n < 1250$ gives $n \leq 277$ , and $5.5 n > 1150$ gives $n \geq 210$ . From $210$ to $277$ there are $68$ integers. Option C forgets to count both ends.

Question 10 · Modelled on 2014 P2 Q11 — a triangle, with statements about which values the area can actually take

The base and the height of a triangle are measured as $10 cm$ and $6 cm$ respectively, correct to the nearest cm. Let $A cm^{2}$ be the actual area of the triangle. Which of the following are true?

I. $A$ can be $27$ .
II. $A$ can be $34.125$ .
III. $A$ and $30$ differ by less than $4.2$ .

I and II only
I and III only
II and III only
I, II and III

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B. $26.125 \leq A < 34.125$ . $27$ is inside. $34.125$ is the upper limit and is never reached. The largest gaps from $30$ are $3.875$ below and just under $4.125$ above.

Continue practising this topic: Go back to the Paper 2 multiple-choice for the standard set, or move to Paper 1 long questions — also available as Paper 1 Challenge.

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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