Ratio, proportion and rates of change worksheet — GCSE Higher

1.The height of water in a tank, h metres, t minutes after a tap is opened is modelled by h = 0.02t² + 0.5. Estimate the instantaneous rate of change of the height at t = 10, using the gradient of the chord joining t = 9 and t = 11.

2.A metal has a density of 7.8 g/cm³. Work out the density of the metal in kg/m³.

3.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.

4.A car is bought for £12000. Its value falls by 15% in the first year and by 10% in each year after that. Work out the value of the car 3 years after it was bought.

5.A car uses fuel at a constant rate. It uses 24 litres of fuel to travel 300 km. Assuming the same rate, work out how many litres of fuel are needed for a journey of 175 km.

6.Before a charity campaign, donations were £6400 per month. Donations grew by 8% in the first year after the campaign, and then fell by 3% in the second year as interest faded. Work out the amount donated per month at the end of the second year, to the nearest pound.

7.The concentration of a pollutant in a lake, in arbitrary units, follows the recurrence C_{n+1} = 0.75C_n + 40, with C_0 = 500: each week, 25% of the pollutant breaks down naturally, and then a further 40 units enter the lake from run-off. An ecologist classifies the lake as safe once C_n first drops below 300. Find the first whole number of weeks after which the lake is safe.

8.It takes 2 identical pumps 10 hours to empty a flooded basement. Working at the same rate, work out how many hours 5 of these pumps would take to empty the same basement.

9.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.

10.Two mathematically similar cylinders have volumes 64 cm³ and 216 cm³. Work out the ratio of the height of the smaller cylinder to the height of the larger cylinder, in simplest form.

11.Maya buys a vintage car for £15000. Its value is expected to increase by 4% each year for 2 years. She then plans to sell the car and use £2000 of the money from the sale to buy a trailer. Work out how much money she will have left after selling the car and buying the trailer.

12.A charity collects donations from adults and children in the ratio 5:2. Altogether, £238 is collected. Work out how much more the adults donate than the children.

13.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.

14.A cyclist rides at a steady speed of 20 mph. Given that 1 mile ≈ 1.6 km, work out the cyclist's speed in km/h.

15.A cyclist rides 19.3 km in 47 minutes. Work out her average speed, in km/h, to 1 decimal place.