Practice › Number and Algebra › Rates, Ratios and Proportions

Rates, Ratios and Proportions: Paper 2 Multiple-Choice

Paper 2 asks about ratios almost every year, usually as a : b : c from two conditions, a map scale, an average speed or a mixture. These ten questions cover each type once or twice, and every answer says which mistake leads to the tempting wrong option. Detailed solutions can be requested by email.

10 questions · about 15 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 2022 P2 Q12 — same shape, with the unknown ratio between the second and third numbers

Let

x

y

and

z

be non-zero numbers. If

x : y = 3 : 4

and

2 y = 5 z - x

, then

y : z =

$11 : 20$ .
$20 : 11$ .
$4 : 11$ .
$15 : 11$ .

Show answer

B. Put $x = 3 k$ , $y = 4 k$ . Then $8 k = 5 z - 3 k$ , so $z = \frac{11 k}{5}$ and $y : z = 4 : \frac{11}{5} = 20 : 11$ . Option D is $x : z$ .

Question 2 · Modelled on 2012 P2 Q9 and 2016 P2 Q11 — one more step: after finding x : y, a second ratio is built from it

x

and

y

are positive numbers such that

(2 x + 3 y) : (x + 2 y) = 8 : 5

, then

(x + y) : (y - x) =

$1 : 2$ .
$2 : 1$ .
$1 : 3$ .
$3 : 1$ .

Show answer

D. $5 (2 x + 3 y) = 8 (x + 2 y)$ gives $y = 2 x$ . Then $(x + y) : (y - x) = 3 x : x = 3 : 1$ . Option A is $x : y$ , the half-way answer.

Question 3 · Modelled on 2021 P2 Q6 and 2024 P2 Q6 — an identity in x, with all three constants asked for

a

b

and

c

are non-zero constants such that

a (x + 3) + b (2 x - 1) \equiv c (x + 1)

, then

a : b : c =

$3 : 2 : 7$ .
$2 : 3 : 8$ .
$3 : 2 : 5$ .
$7 : 2 : 3$ .

Show answer

A. Comparing coefficients: $a + 2 b = c$ and $3 a - b = c$ . So $a + 2 b = 3 a - b$ , giving $3 b = 2 a$ . With $a = 3$ , $b = 2$ : $c = 7$ . Option B satisfies the first equation only.

Question 4 · Modelled on 2013 P2 Q12 — same direction (find the scale from two areas), with the actual area in km²

The actual area of a lake is

1.44 km^{2}

. If the area of the lake on a map is

9 cm^{2}

, then the scale of the map is

$1 : 4000$ .
$1 : 16000$ .
$1 : 40000$ .
$1 : 160000$ .

Show answer

C. $1 cm^{2}$ stands for $1.44 \div 9 = 0.16 km^{2}$ , so $1 cm$ stands for $0.16 = 0.4 km = 40000 cm$ . Option B forgets the square root.

Question 5 · Modelled on 2017 P2 Q11 — the region on the map is given by its two sides, not its area

The scale of a map is

1 : 5000

. A rectangular field measures

3 cm

4 cm

on the map. The actual area of the field is

$6 \times 1 0^{2} m^{2}$ .
$3 \times 1 0^{4} m^{2}$ .
$6 \times 1 0^{4} m^{2}$ .
$3 \times 1 0^{8} m^{2}$ .

Show answer

B. $1 cm$ stands for $5000 cm = 50 m$ . The field is $150 m$ by $200 m$ , so its area is $30000 m^{2}$ . Option D is the same number left in $cm^{2}$ .

Question 6 · Modelled on 2024 P2 Q12 — two stages at different speeds for different times

Mandy walks at an average speed of

6 km/h

for

20

minutes and then cycles at an average speed of

18 km/h

for

40

minutes. Her average speed for the whole journey is

$10 km/h$ .
$12 km/h$ .
$14 km/h$ .
$16 km/h$ .

Show answer

C. Distance $= 6 \times \frac{1}{3} + 18 \times \frac{2}{3} = 2 + 12 = 14 km$ in exactly $1$ hour. Option B is the plain average of $6$ and $18$ , which ignores the unequal times.

Question 7 · Modelled on 2012 P2 Q11 — run backwards: the overall average rate is given and one stage rate is the unknown

A machine prints at an average rate of

45

pages per minute for the first

4

minutes and at an average rate of

r

pages per minute for the next

6

minutes. If its average rate for the

10

minutes is

54

pages per minute, then

r =

$60$ .
$63$ .
$66$ .
$72$ .

Show answer

A. Total pages $= 54 \times 10 = 540$ . The first stage gives $45 \times 4 = 180$ , leaving $360$ pages in $6$ minutes. Option B treats $54$ as the midpoint of $45$ and $r$ .

Question 8 · Modelled on 2015 P2 Q11 — the joined ratio is used to share a sum of money

A sum of

$700

is shared among

P

Q

and

R

. If the ratio of the share of

P

to the share of

Q

2 : 3

and the ratio of the share of

Q

to the share of

R

4 : 5

, then the share of

R

$$160$ .
$$240$ .
$$280$ .
$$300$ .

Show answer

D. $P : Q : R = 8 : 12 : 15$ , which is $35$ parts of $$20$ each. $R$ gets $15 \times 20$ . Options A and B are the shares of $P$ and $Q$ .

Question 9 · Modelled on 2019 P1 Q7 — only one group changes, and the total is asked for

In a club, the ratio of the number of boys to the number of girls is

5 : 4

. If

6

boys leave the club, then the number of boys and the number of girls are equal. Find the original number of members in the club.

$48$
$54$
$60$
$72$

Show answer

B. With $5 k$ boys and $4 k$ girls, $5 k - 6 = 4 k$ gives $k = 6$ , so there were $9 k = 54$ members. Option A is the number left after the boys go.

Question 10 · Modelled on 2019 P2 Q12 and 2026 P2 Q10 — neither price is given; two mixtures have to be compared to find one

Tea of brand

A

and tea of brand

B

are mixed. If they are mixed in the ratio

3 : 2

by weight, the cost of the mixture is

$114/ kg

. If they are mixed in the ratio

2 : 3

by weight, the cost of the mixture is

$126/ kg

. Find the cost of tea of brand

A

$$60/ kg$
$$84/ kg$
$$90/ kg$
$$150/ kg$

Show answer

C. With costs $a$ and $b$ : $3 a + 2 b = 570$ and $2 a + 3 b = 630$ . Adding gives $a + b = 240$ ; subtracting gives $b - a = 60$ . So $a = 90$ . Option D is brand $B$ .

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

Where this sits in the syllabus

Rates, ratios and proportions is a junior secondary topic that the HKDSE keeps testing. Between 2012 and 2026 most Paper 2 papers carry a question on a : b : c, a map scale, an average speed or a mixture, and Paper 1 Section A(1) uses the topic for short word problems. The k-method practised here returns later in similar figures, variations and trigonometric ratios.

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

Start with a free discovery session

No cost, no obligation. We look at where your child actually is, and show you what a term of real progress would look like.

Book a free discovery session See the programmes