24 questions across ratio, direct and inverse proportion, compound measures and area and volume scale factors.

⚗️ Ratio and proportion mastery — Higher

Maths · GCSE Higher · 24 questions · ~45 min

MathsUKwww.geekhero.co.uk

Ratio, proportion and rates of change is 20% of a Higher paper and it is the area that rewards method over recall. This sheet works through it in one sitting: simplifying and sharing in a ratio, including the questions that give you one share or the difference rather than the total; expressing one quantity as a fraction or percentage of another; direct and inverse proportion, both numerically and as an equation with a constant k; compound measures — speed, density and pressure — with unit conversions built in rather than avoided; and the similar-shapes work where lengths, areas and volumes scale by the factor, its square and its cube. That last group is where Higher candidates lose the most marks in this area, so it is deliberately over-represented here.

1.A concrete block exerts a force of 45 N on the ground through a base with an area of 0.05 m². Work out the pressure on the ground, in N/m².
(a)9 N/m²
(b)900 N/m²
(c)45.05 N/m²
(d)2.25 N/m²
2.A statue exerts a downward force of 68.4 N on its base, which has an area of 2.85 m². Work out the pressure that the statue exerts on its base, in N/m².
(a)24 N/m²
(b)194.94 N/m²
(c)65.55 N/m²
(d)0.04 N/m²
3.The density of a type of wood is 0.8 g/cm³. Work out the mass of a piece of this wood with a volume of 150 cm³.
(a)149.2 g
(b)187.5 g
(c)1200 g
(d)120 g
4.y is inversely proportional to x². When x = 2, y = 8. Construct the equation connecting x and y, then work out the value of y when x = 4.
(a)2
(b)512
(c)4
(d)8
5.An amount of money is shared in the ratio 1:2:3. The largest share is £90 more than the smallest share. Work out the total amount that was shared.
(a)£180
(b)£540
(c)£135
(d)£270
6.Two mathematically similar tins have heights 8 cm and 12 cm. Jayden says that the volume of the larger tin is 1.5 times the volume of the smaller tin. Give a reason why Jayden is incorrect, and work out the correct volume scale factor.
(a)3.375
(b)2.25
(c)4.5
(d)1.5
7.y is directly proportional to x², and x is positive. When x = 4, y = 32. Construct the equation connecting x and y, then work out the value of x when y = 200.
(a)10
(b)25
(c)100
(d)20
8.An architect builds a scale model of a staircase before construction. The model is mathematically similar to the real staircase, at a scale of 1 : 10. The model covers a floor area of 0.4 m² on the plan. Work out the floor area the real staircase will cover, giving your answer in m².
(a)40 m²
(b)0.04 m²
(c)400 m²
(d)4 m²
9.Two cars leave the same point at the same time and travel in opposite directions. One travels at 80 km/h and the other at 60 km/h. Work out how long it takes until the cars are 350 km apart.
(a)5 hours
(b)2.5 hours
(c)17.5 hours
(d)4.375 hours
10.Triangle ABC is mathematically similar to triangle PQR, with AB corresponding to PQ and BC corresponding to QR. AB = 6 cm, BC = 8 cm and PQ = 12 cm. Work out the length of QR.
(a)32 cm
(b)24 cm
(c)16 cm
(d)14 cm
11.Tap A fills a swimming pool in 6 hours. Tap B pours water twice as fast as tap A. The time taken to fill the pool is inversely proportional to the rate of flow. Work out how long tap B takes to fill the pool.
(a)3 hours
(b)12 hours
(c)4 hours
(d)1.5 hours
12.The cost of manufacturing a spherical container is proportional to the cube of its radius. A container of radius 3 cm costs £54 to manufacture. Construct the equation connecting cost C and radius r, then work out the cost of a container of radius 5 cm.
(a)£150
(b)£250
(c)£10
(d)£90
13.Write 400 g : 1.5 kg as a ratio in its simplest form.
(a)4 : 15
(b)40 : 150
(c)4 : 1.5
(d)15 : 4
14.A recipe uses flour, sugar and butter in the ratio 8 : 3 : 5. Write the ratio of flour to the rest of the mixture (sugar and butter combined) in its simplest form.
(a)3 : 5
(b)8 : 3
(c)1 : 2
(d)1 : 1
15.y is inversely proportional to x², and x is positive. When x = 2, y = 45. Construct the equation connecting x and y, then work out the value of x when y = 5.
(a)6
(b)36
(c)18
(d)30
16.Two mathematically similar jugs have heights 8 cm and 12 cm. The smaller jug holds 200 ml when it is full. Work out how much the larger jug holds when it is full.
(a)675 ml
(b)300 ml
(c)450 ml
(d)900 ml
17.Write 75p : £1.50 as a ratio of whole numbers in its simplest form.
(a)2:1
(b)3:4
(c)1:2
(d)50:1
18.Noah runs 2 km in 10 minutes. Work out how long he takes to run 5 km at the same speed.
(a)50 minutes
(b)20 minutes
(c)15 minutes
(d)25 minutes
19.Ffion is paid £11.20 per hour. She works 6 hours on Monday and 4.5 hours on Tuesday. Work out her total pay for the two days.
(a)£67.20
(b)£50.40
(c)£106.40
(d)£117.60
20.Priya is paid £58.50 for working 7.5 hours on a Saturday. Work out her rate of pay, in £ per hour.
(a)£7.80
(b)£438.75
(c)£51.00
(d)£0.13
21.A recipe for 8 muffins needs 200 g of sugar. Sam wants to make 20 muffins for a bake sale, and he already has 350 g of sugar. Work out how many more grams of sugar he needs to buy.
(a)150 g
(b)850 g
(c)0 g
(d)500 g
22.A coach travels the 54 miles from London to Brighton in 1 hour 30 minutes. Work out the average speed of the coach, in mph.
(a)1.8 mph
(b)41.54 mph
(c)81 mph
(d)36 mph
23.A delivery driver travels 45 km in the first 30 minutes of a journey, and then a further 75 km in the next 1 hour. Work out her average speed for the whole journey, in km/h.
(a)80 km/h
(b)75 km/h
(c)82.5 km/h
(d)60 km/h
24.y is directly proportional to x². When x = 3, y = 45. Construct the equation connecting x and y, then work out the value of y when x = 5.
(a)50
(b)25
(c)75
(d)125

What is on this worksheet?

The sheet holds 24 questions drawn from the MathsUK bank — the content area covered: Ratio, proportion and rates of change (statements R4, R5, R9, R10, R11, R12, R13). It is pitched at GCSE Higher and takes about 45 minutes to work through in full. It is built for independent practice, with full answers at the end for self-marking.

How to use the sheet well

Print it or open it on screen — both work. Printing is A4; the screen view fits phones and tablets.
Do all 24 questions before checking — about 45 minutes is the guide, but there is no time pressure.
Check the answers — press “Show answers” or print the answer page separately.
Redo the questions you got wrong — twice as effective as doing 24 fresh ones.
“New questions” — builds a fresh sheet on the same statements, so you can practise again without repeats.

Why this sheet helps

MathsUK worksheets use questions graded by difficulty and a fair spread of correct-answer positions (the answer is not always (a)) — so the student really has to think about each question rather than guess a pattern. Every question is tagged to a DfE content statement and checked before it enters the bank. The answers come with a step-by-step explanation, not just a value — so a wrong answer becomes a lesson.

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