Ratio, proportion and rates of change worksheet — GCSE Higher

1.A candidate wants to estimate the instantaneous rate of change of a reservoir's water level, in metres, at t = 5 days after heavy rain began. The reservoir's water level is plotted against time, in days, since the rain began. The candidate uses the chord joining the points at t = 0 days and t = 20 days to estimate the rate of change at t = 5 days. Give a reason why this is likely to be a poor estimate.

2.A curve has equation y = 2x² + 5. Work out an estimate for the gradient of the curve at x = 3, using a chord between the points where x = 2 and x = 4.

3.The tangent to a curve at the point where x = 4 has equation y = 3x − 2. Work out the instantaneous rate of change of y with respect to x at x = 4.

4.The ratio of Josh's savings to Mia's savings is 4:7. Mia has £39 more than Josh. Work out Josh's savings.

5.A shade of paint is made by mixing blue paint and white paint. To make 5 litres of the shade, 2 litres of blue paint is used and the rest is white paint. Write the ratio of blue paint to white paint in its simplest form.

6.To make a shade of green paint, the amount of yellow paint used is always 0.75 times the amount of blue paint used. Write the amount of blue paint as a fraction of the amount of yellow paint, in its simplest form.

7.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.

8.A water tank has a capacity of 500 litres. It currently holds 350 litres of water. Write the amount of empty space in the tank as a fraction of the tank's capacity. Give your answer in its simplest form.

9.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.

10.A recipe for 6 muffins needs 150 g of flour. Aisha wants to make 15 muffins using the same recipe. Work out how much flour she needs.

11.A jug holds 3 litres of a drink that is 60% fruit juice. 1 litre of water is added to the jug. Work out the percentage of the new mixture that is fruit juice.

12.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.

13.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 Each year, 4% interest is added, and then £30 is withdrawn from the account.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?

14.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.

15.The ratio of the amount Noah has saved to the amount Grace has saved is 4:5. Noah has saved £200. Work out how much they have saved altogether.