Ratio, proportion and rates of change worksheet — GCSE Higher

1.A tank is filled at a rate of 3 litres every 2 minutes. Given that 1 litre = 1000 cm³, work out the flow rate in cm³ per minute.

2.The population of a town, P thousand, is plotted against time, t years, since 2020. A tangent to the graph at t = 6 has gradient 2.5. What does this gradient represent?

3.A curve has equation y = x² + 2x. Work out the average rate of change of y with respect to x over the interval from x = 1 to x = 4.

4.A tap fills a tank at a rate of 15 litres per minute. Given that 1 litre = 1000 cm³, work out the rate at which the tank fills in cm³ per second.

5.A savings account starts with £500. Each year, 4% interest is added, and then £30 is withdrawn from the account. Which recurrence correctly models the balance, £S_n, after n years, with S_0 = 500?

6.A conversion graph between kilometres and miles passes through the origin and the point (50, 31), with the number of kilometres on the horizontal axis and the number of miles on the vertical axis. Work out the number of miles equivalent to 1 kilometre.

7.A metal sample has a mass of 342.6 g and a volume of 18 cm³. Work out the density of the sample, in g/cm³, to 1 decimal place.

8.Leah puts £4000 into a savings account paying 3% compound interest each year. At the end of 2 years she takes out all of the money and spends £1500 of it on a laptop. Work out how much of the money she has left.

9.y is directly proportional to √x. When x = 25, y = 20. Construct the equation connecting x and y, then work out the value of x when y = 32.

10.6 identical taps fill a paddling pool in 20 minutes. Each tap fills at the same steady rate. Work out how long 3 of these taps would take to fill the same pool.

11.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.

12.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.

13.A sculptor makes two mathematically similar statues. The smaller statue is 20 cm tall and 80 ml of varnish covers its surface. The larger statue is 50 cm tall. Work out how much varnish is needed to cover the surface of the larger statue.

14.A jug of squash is made by mixing water and syrup in the ratio 6:1. Nia wants to make 8.4 litres of squash. Work out how much syrup she needs, in litres.

15.A coach journey is 372 miles in total. After a stop, the coach has travelled 217 miles. Write the distance still to travel as a fraction of the total journey. Give your answer in its simplest form.