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Rates, Ratios and Proportions: Paper 1 Long Questions

A ratio tells you how quantities compare, never how big they are. These ten questions practise turning a ratio into a single unknown k, then move on to map scales, average speed, work rates, a travel graph and mixtures. Detailed solutions can be requested by email.

10 questions · 47 marks · about 55 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 2025 P1 Q3 and 2016 P2 Q11 — the ratio of the two numbers is not given; it has to be recovered from a ratio of their sum and difference first

Let $p$ and $q$ be two positive numbers. The ratio of $p + q$ to $p - q$ is $9 : 1$ while $3 p - 2 q = 84$ . Find $q$ . (3 marks)

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$q = 48$ . From $p + q = 9 (p - q)$ we get $p : q = 5 : 4$ . Put $p = 5 k$ , $q = 4 k$ : then $7 k = 84$ , so $k = 12$ .

Question 2 · Modelled on 2020 P1 Q4 and 2018 P2 Q10 — one link is given as an equation, and the three-term ratio is asked for before the fraction

Let $a$ , $b$ and $c$ be non-zero numbers such that $a : b = 3 : 4$ and $5 b = 6 c$ .

(a) Find $a : b : c$ . (2)

(b) Find the value of $\frac{3 a - c}{a + b + c}$ . (2)

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(a) $a : b : c = 9 : 12 : 10$ . $5 b = 6 c$ means $b : c = 6 : 5$ ; scale both ratios so that $b$ is $12$ .
(b) $\frac{17}{31}$ . With $a = 9 k$ , $b = 12 k$ , $c = 10 k$ the $k$ cancels.

Question 3 · Modelled on 2019 P1 Q7 — instead of people arriving from outside, marbles move between the two groups, so the total stays fixed

Box $A$ and box $B$ contain some marbles. The ratio of the number of marbles in box $A$ to the number of marbles in box $B$ is $7 : 3$ . If $18$ marbles are moved from box $A$ to box $B$ , then the ratio becomes $2 : 3$ . Find the total number of marbles in the two boxes. (4 marks)

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$60$ marbles. With $7 k$ and $3 k$ marbles, $\frac{7 k - 18}{3 k + 18} = \frac{2}{3}$ gives $k = 6$ . Check: $24 : 36 = 2 : 3$ .

Question 4 · Modelled on 2017 P2 Q11 and 2023 P2 Q12 — both directions in one question: map to ground for a length, ground to map for an area

The scale of a map is $1 : 25000$ .

(a) The distance between two piers on the map is $6.4 cm$ . Find the actual distance between the two piers in km. (2)

(b) The actual area of a country park is $5 km^{2}$ . Find the area of the country park on the map in $cm^{2}$ . (2)

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(a) $1.6 km$ . $6.4 \times 25000 = 160000 cm$ .
(b) $80 cm^{2}$ . $1 cm$ stands for $0.25 km$ , so $1 cm^{2}$ stands for $0.2 5^{2} = 0.0625 km^{2}$ , and $5 \div 0.0625 = 80$ .

Question 5 · Modelled on 2024 P2 Q12 — run backwards: the average speed is given and the time of the second stage is the unknown

A cyclist rides at an average speed of $24 km/h$ for $40$ minutes and then rides at an average speed of $15 km/h$ for $t$ minutes. The average speed of the cyclist for the whole journey is $18 km/h$ . Find $t$ . (4 marks)

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$t = 80$ . Total distance $= 16 + \frac{t}{4}$ km and total time $= \frac{40 + t}{60}$ h. Check: $36 km$ in $2$ hours is $18 km/h$ .

Question 6 · Modelled on 2018 P1 Q9 — the two stages now cover the same road, so the link between them is equal distance, not a known total distance

A van travels from town $A$ to town $B$ at an average speed of $60 km/h$ and then returns from town $B$ to town $A$ along the same road at an average speed of $40 km/h$ . The van spends $3$ hours $20$ minutes travelling in total.

(a) Find the distance between town $A$ and town $B$ . (3)

(b) Someone claims that the average speed of the van for the whole trip is $50 km/h$ . Do you agree? Explain your answer. (2)

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(a) $80 km$ . $\frac{d}{60} + \frac{d}{40} = \frac{10}{3}$ .
(b) Disagree. The average speed is $160 \div \frac{10}{3} = 48 km/h$ : the van spends longer at the slower speed, so the average sits below $50$ .

Question 7 · Modelled on 2022 P2 Q12 and 2015 P2 Q11 — the joined ratio is put to work in a sharing problem where only a difference is known

A sum of money is shared among Amy, Ben and Carl. The ratio of Amy's share to Ben's share is $4 : 3$ and the ratio of Ben's share to Carl's share is $2 : 5$ .

(a) Find Amy's share : Ben's share : Carl's share. (2)

(b) Carl's share is $$196$ more than Amy's share. Find the sum of money. (3)

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(a) $8 : 6 : 15$ . Make Ben $6$ in both ratios.
(b) $$812$ . $15 k - 8 k = 196$ gives $k = 28$ , and the sum is $29 k$ .

Question 8 · Modelled on 2012 P2 Q11 (a rate that is not a speed) — two rates now work at the same time, then one stops

Pipe $A$ alone can fill an empty tank in $12$ hours. Pipe $B$ alone can fill the same empty tank in $20$ hours.

(a) If both pipes are turned on together, how long does it take to fill the empty tank? (2)

(b) Both pipes are turned on together to fill the empty tank. After $3$ hours, pipe $A$ is turned off. How much longer does pipe $B$ take to fill the tank? (3)

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(a) $7.5$ hours. The rates add: $\frac{1}{12} + \frac{1}{20} = \frac{2}{15}$ of the tank per hour.
(b) $12$ hours. After $3$ hours $\frac{2}{5}$ is filled; $B$ fills the other $\frac{3}{5}$ at $\frac{1}{20}$ per hour.

Question 9 · Modelled on 2014 P1 Q10 — the cars now travel towards each other, and they meet while one of them is at rest

Town $P$ and town $Q$ are $90 km$ apart. The figure shows the graphs for car $A$ and car $B$ travelling on the same straight road between town $P$ and town $Q$ . Car $A$ leaves town $P$ at 8:00 and travels at a constant speed to town $Q$ , arriving at 9:30. Car $B$ leaves town $Q$ at 8:00, comes to rest at 8:36 at a place $54 km$ from town $P$ , starts again at 9:00 and reaches town $P$ at 9:45.

Travel graph: distance from town P against time for car A and car B

(a) Find the distance of car $A$ from town $P$ at 8:36. (2)

(b) At what time do car $A$ and car $B$ meet? (2)

(c) The driver of car $B$ claims that the average speed of car $B$ from 8:00 to 9:45 is higher than the speed of car $A$ . Do you agree? Explain your answer. (2)

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(a) $36 km$ . Car $A$ travels at $90 \div 1.5 = 60 km/h$ , and $36$ minutes is $0.6$ h.
(b) 8:54. At 8:36 the cars have not yet met ( $36 km$ against $54 km$ ); car $B$ then waits at $54 km$ , and car $A$ gets there after $54$ minutes.
(c) Disagree. Car $B$ covers $90 km$ in $1.75$ h, about $51.4 km/h$ , which is less than $60 km/h$ .

Question 10 · Modelled on 2019 P2 Q12 and 2026 P2 Q10 — the mixture question grown into a long question: cost from ratio, ratio from cost, then changing one mixture into the other

The costs of coffee beans of brand $X$ and brand $Y$ are $$80/ kg$ and $$120/ kg$ respectively.

(a) Coffee beans of brand $X$ and brand $Y$ are mixed in the ratio $3 : 2$ by weight. Find the cost of the mixture per kg. (2)

(b) If $x kg$ of brand $X$ and $y kg$ of brand $Y$ are mixed so that the cost of the mixture is $$104/ kg$ , find $x : y$ . (2)

(c) A shopkeeper has $12 kg$ of the mixture described in (a). How many kg of brand $Y$ should be added so that the cost of the new mixture is $$104/ kg$ ? (3)

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(a) $$96/ kg$ . $\frac{3 ( 80 ) + 2 ( 120 )}{5} = 96$ .
(b) $x : y = 2 : 3$ . $80 x + 120 y = 104 (x + y)$ gives $16 y = 24 x$ .
(c) $6 kg$ . The $12 kg$ holds $7.2 kg$ of $X$ and $4.8 kg$ of $Y$ ; for $2 : 3$ the $Y$ must rise to $10.8 kg$ .

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

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.

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