Ratio, proportion and rates of change worksheet — GCSE Higher

1.A garden centre sells two mathematically similar sacks of grass seed. The amount of lawn a sack can treat is proportional to the volume of seed inside it. The smaller sack is 20 cm tall and treats a lawn of area 30 m². The larger sack is 40 cm tall. A gardener needs to treat a lawn with an area of 500 m² using only the larger sacks. Work out the minimum number of larger sacks needed.

2.A company's turnover was £128 000 in 2023 and £176 000 in 2024. Work out the multiplier that changes the 2023 turnover into the 2024 turnover.

3.A furniture maker builds two mathematically similar bookcases. The smaller bookcase is 40 cm tall and uses 2 m² of wood. The larger bookcase is 60 cm tall. Work out the area of wood needed for the larger bookcase, giving your answer in m².

4.In a class, 50% of the students study French, 30% study Spanish and the rest study German. Write down the ratio of French : Spanish : German students, in its simplest form.

5.A lorry is carrying a load of mass m tonnes. Given that 1 tonne = 1000 kg, write down an expression for the mass of the load in kilograms.

6.Two straight-line graphs represent y = 2x and y = 5x. For the same value of x, where x ≠ 0, write down the ratio of the y-value on the first graph to the y-value on the second graph, in its simplest form.

7.Two quantities, P and Q, are in inverse proportion. A graph is drawn with P on the vertical axis and Q on the horizontal axis. Write down which description fits the shape of this graph.

8.Two mathematically similar cubes have edge lengths 2 cm and 6 cm. Write the ratio of the volume of the smaller cube to the volume of the larger cube in its simplest form.

9.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.

10.A laptop costs £800 when new. Its value decreases by 25% of its value at the start of each year. Work out how much value the laptop loses in the second year.

11.After a 20% discount, a jacket costs £48. Work out the original price of the jacket.

12.The price of a jacket increases by 50% and then decreases by 50%. Describe the overall change from the original price.

13.The number of subscribers to a streaming app, in thousands, is plotted against time, in months since launch. A tangent to the graph at t = 6 months has gradient 4.8. A tangent at t = 18 months has gradient 1.1. A manager claims the app is growing faster at 18 months than it was at 6 months. Work out how the growth rate has changed, and decide whether the manager is correct.

14.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.

15.A cyclist travels 18 km in 45 minutes. Work out the average speed, in km/h.