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 metal cylinder has a mass of 356.5 g and a volume of 47 cm³. Work out the density of the cylinder, in g/cm³, to 1 decimal place.
(a)0.1
(b)403.5
(c)7.6
(d)7.5
2.Maya is paid £105.30 for working 9 hours. Work out her rate of pay, in £ per hour.
(a)£11.70
(b)£117.00
(c)£947.70
(d)£96.30
3.A factory machine produces bottles at a constant rate. In 45 minutes it produces 810 bottles. The factory needs 2,160 bottles for an order. Working at the same rate, work out how many minutes it will take to produce the order.
(a)48 minutes
(b)120 minutes
(c)108 minutes
(d)1,350 minutes
4.The number of members of a running club increases from 45 to 54. Work out the percentage increase.
(a)9%
(b)120%
(c)20%
(d)16.7%
5.A shop sells rope by the metre. 3 m costs £7.50 and 5 m costs £11.00. Does this data show that the cost is directly proportional to the length of rope bought? Choose the correct verdict and reason.
(a)No — the cost per metre differs: £2.50/m vs £2.20/m
(b)Yes — because both amounts increase as length increases
(c)No — because 5 m costs more than 3 m in total
(d)Yes — the cost per metre is £2.50 in both cases
6.Two mathematically similar hexagonal tiles have areas 18 cm² and 50 cm². Work out the ratio of the side length of the smaller tile to the side length of the larger tile, in simplest form.
(a)9 : 25
(b)5 : 3
(c)18 : 50
(d)3 : 5
7.A spring's extension is directly proportional to the force applied to it. A force of 5 N produces an extension of 12 mm. Work out the extension produced by a force of 20 N.
(a)32 mm
(b)3 mm
(c)48 mm
(d)36 mm
8.The number of euros, e, received is directly proportional to the number of pounds, p, exchanged. Exchanging £40 gives 46 euros. Work out how many euros are received for £65, giving your answer to the nearest euro.
(a)57
(b)75
(c)74
(d)71
9.A company's turnover this year is £180,000. Last year's turnover was £120,000. Write down this year's turnover as a percentage of last year's turnover.
(a)50%
(b)180%
(c)150%
(d)66.7%
10.A 1.5 kg bag of pasta costs £2.85 and a 2.4 kg bag of the same pasta costs £4.32. Work out which bag gives better value for money.
(a)The 1.5 kg bag, since it costs £1.19 per kg compared with £2.88 per kg for the 2.4 kg bag.
(b)The 2.4 kg bag, since it costs £1.80 per kg compared with £1.90 per kg for the 1.5 kg bag.
(c)The 1.5 kg bag, since it costs £1.80 per kg compared with £1.90 per kg for the 2.4 kg bag.
(d)The 1.5 kg bag, since £2.85 is less than £4.32.
11.A machine fills bottles at a constant rate. It fills 18 bottles in 3 minutes. Working at the same rate, work out how many bottles the machine fills in 8 minutes.
(a)144
(b)18
(c)23
(d)48
12.y is directly proportional to √x. When x = 4, y = 8. Construct the equation connecting x and y, then work out the value of y when x = 9.
(a)18
(b)12
(c)36
(d)3
13.2.4 kg of cheese costs £36. Work out the cost of 1.5 kg of the same cheese.
(a)£15
(b)£57.60
(c)£22.50
(d)£54
14.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
15.A printer prints at a constant rate. The time taken to print a batch of forms is inversely proportional to the printer's speed, in pages per minute. Printing at 20 pages per minute takes 15 minutes. Work out how long the same batch takes to print at 25 pages per minute. Give your answer in minutes.
(a)12
(b)10
(c)18.75
(d)20
16.Write 2 m : 150 cm : 50 cm as a ratio of whole numbers in its simplest form.
(a)4:3:1
(b)4:3:2
(c)2:150:50
(d)3:4:1
17.The number of tickets a group can afford is inversely proportional to the price per ticket. At £4 per ticket, the group can afford 12 tickets. Work out how many tickets the group can afford at £6 per ticket.
(a)6
(b)18
(c)12
(d)8
18.Write 250 g : 2 kg as a ratio in its simplest form.
(a)1:8
(b)5:40
(c)125:1
(d)8:1
19.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
20.The width, the length and the height of a box are in the ratio 3:4:5. The length of the box is 16 cm. Work out the height of the box.
(a)20 cm
(b)80 cm
(c)4 cm
(d)12 cm
21.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
22.The manufacturing cost of a specialist lens, in pounds, is proportional to the square root of its diameter, in millimetres. A lens of diameter 9 mm costs £12 to manufacture. Construct the equation connecting cost C and diameter d, then work out the cost of manufacturing a lens of diameter 16 mm.
(a)£32
(b)£16
(c)£4
(d)£64
23.Which of these ratios is equivalent to 6 : 10 : 14?
(a)8:12:16
(b)3:5:8
(c)9:15:21
(d)12:20:14
24.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

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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